On Existence and Regularity for a Cahn-Hilliard Variational Model for Lithium-Ion Batteries
aa r X i v : . [ m a t h . A P ] O c t On Existence and Regularity for a Cahn-HilliardVariational Model for Lithium-Ion Batteries
Kerrek [email protected] Mellon UniversityOctober 19, 2020
Abstract
The Cahn-Hilliard reaction model, a nonlinear, evolutionary system of PDEs, was in-troduced to model phase separation in lithium-ion batteries. Using Butler-Volmer kineticsfor electrochemical consistency, this model incorporates a nonlinear Neumann boundarycondition ∂ ν µ = R ( c, µ ) for the chemical potential µ , with c , the lithium-ion density. Im-portantly, R depends exponentially on µ . In arbitrary dimension, existence of a weaksolution for the Cahn-Hilliard reaction model with elasticity is proven using a generalizedgradient structure. This approach is, at present, restricted to polynomial growth in R .Working to remove this limitation, fixed point methods are applied to obtain existenceof strong solutions of the Cahn-Hilliard reaction model without elasticity in dimensions 2and 3 . This method is then extended to prove existence of higher regularity solutions indimension 2, allowing for recovery of exponential boundary conditions as in the physicalapplication to lithium-ion batteries.
Key words:
Cahn-Hilliard, gradient flows, lithium-ion batteries
AMS Classifications:
Introduction
Modern technology relies heavily on lithium-ion batteries, from mobile phones to hybrid cars.More broadly, for the use of inconsistent renewable energies such as solar, it is imperative todevelop effective means of energy storage, and lithium-ion batteries are premiere candidates forsuch storage [1]. The centrality of the need for a better understanding of batteries was under-scored by the 2019 Nobel Prize in Chemistry, which was awarded to Goodenough, Whittingham,and Yoshino for their pioneering works in the development of lithium-ion batteries [1].A prominent phenomenological behavior of lithium-ion batteries is phase separation, whereinlithium-ions intercalate into the host structure of the cathode inhomogeneously. In contrast toclassical fluid-fluid transitions, such as oil and water, the separation of lithium-ions takes placewithin a solid host. Consequently, phase transitions induce a strain, damaging the cathode’shost material, and this leads to a decrease in battery performance and limited life-cycle (see [9],[23], and references therein).Understanding the onset of phase transitions is, therefore, imperative to improving batteryperformance, and much work has been done in this direction. Contemporary paradigms formodeling lithium-ion batteries are moving towards the incorporation of phase field models, alsoknown as diffuse interface models (see, e.g., [6], [8], [21], [52], [58]). These phase field modelsare governed by global energy functionals, which have regular inputs (e.g., Sobolev functions).As noted in [9], the phase field field model is robust, allowing for electrochemically consistentmodels for the time evolution of lithium-ion batteries. Competing models include the shrinkingcore model and the sharp interface model. As noted in Burch et al. [15], the shrinking coremodel fails to capture fundamental qualitative behavior. Furthermore, in [40] it is proposedthat the phase field model may provide a more accurate numerical analysis of the problem thanthe sharp interface model, which seeks to model the evolution of the phase boundary as a freeboundary problem (see [16]; see also [3], and references therein, for benefits of the phase fieldmodel).In this paper we study a variational model introduced by Singh et al. in [58], and incorpo-rating elasticity as proposed by Cogswell and Bazant in [21] (see also [9], [14], [60]), to studythe evolution of a crystal of the battery’s cathode material, such as LiFePO . For a fixed do-main Ω ⊂ R N , the free energy functional associated with the phase field model is given by theCahn-Hilliard energy coupled with linearized elasticity, as introduced by Cahn and Larch´e [43], I el [ u, c ] := Z Ω (cid:18) f ( c ) + ρ k∇ c k + 12 C ( e ( u ) − ce ) : ( e ( u ) − ce ) (cid:19) dx. (1.1)Here c : Ω → [0 ,
1] stands for the normalized density of lithium-ions, u : Ω → R N representsthe material displacement with symmetrized gradient e ( u ) := ∇ u + ∇ u T , e ∈ R N × N is the latticemisfit, ρ > C is a symmetric, positive definite, fourth ordertensor that captures the material constants (stiffness) and satisfies C : R N × N → R N × N sym , C ( ξ ) : ξ > ξ ∈ R N × N sym with ξ = 0 . (1.2)In the cases studied by Bazant et al. (see, e.g., [21], [58]), f is the physically relevant regularsolution free energy with f ( s ) := ωs (1 − s ) + KT abs ( s log( s ) + (1 − s ) log(1 − s )) , s ∈ [0 , , (1.3)where ω ∈ R is a regular solution parameter (enthalpy of mixing), K > T abs > µ is givenby the first variation with respect to c of the energy potential, µ := δ c I el = − ρ ∆ c + f ′ ( c ) + C ( ce − e ( u )) : e . (1.4)1ssuming a quasi-static equilibrium, Singh et al., in [58], and Cogswell and Bazant, in [21], useButler-Volmer kinetics to derive the electrochemically consistent model for the evolution of acrystal of the cathode material given by ∂ t c = div(M(c) ∇ µ ) in Ω , div( C (e(u) − ce )) = 0 in Ω ,∂ ν c = 0 on Γ , ( M ( c ) ∇ µ ) · ν = R ( c, µ ) on Γ , C ( e ( u ) − ce ) ν = 0 on Γ , (1.5)where Γ := ∂ Ω , M is the mobility tensor (degenerate at 0), and R ( s, w ) := R ins − R ext = k ins exp( β ( µ e − w )) − k ext s exp( β ( w − µ e )) , s ∈ (0 , , w ∈ R , (1.6)with constant µ e and positive constants k ext , k ins , β . The first equality of (1.6) emphasizesthat R is a reaction rate determined by the insertion rate R ins minus the extraction rate R ext oflithium-ions. The system of PDEs (1.5) is referred to as the Cahn-Hilliard reaction model or theCHR model. Furthermore, looking to the classical free energy proposed by Cahn and Hilliard[17], I [ c ] := Z Ω (cid:16) f ( c ) + ρ k∇ c k (cid:17) dx, (1.7)where ρ and f are as in (1.1), we may define µ as the first variation of (1.7) to consider (1.5)without elasticity as was first done by Singh et al. [58]. The primary purpose of this paper isto examine existence of solutions of the system of PDEs (1.5) with and without elasticity. Themethods developed in this paper are inspired by the vast literature on the subject.In 1958, Cahn and Hilliard proposed the free energy (1.7) to model isotropic systems ofvarying density [17]. Considering the mass balance equation of the free energy (1.7) in contextof a constitutive equation similar to Fick’s law, one obtains the equation ∂ t c = − div( h ) , h = − M ∇ µ, (1.8)where M is a mobility function and µ is as before the first variation of (1.7) [49]. In manyapplications, M is dependent on c and degenerate at the wells of f . Thermodynamic consistencyand conservation of mass require equation (1.8) be equipped with Neumann boundary conditions ∂ ν c = 0 and ( M ( c ) ∇ µ ) · ν = 0 , respectively. Altogether, we have the Cahn-Hilliard equation ∂ t c = div( M ( c ) ∇ µ ) in Ω ,∂ ν c = 0 on Γ , ( M ( c ) ∇ µ ) · ν = 0 on Γ . (1.9)Many works on the Cahn-Hilliard equation (1.9) make simplifying assumptions dependent on themotivation. In 1986, Elliott and Somgmu [28] used Galerkin methods and a priori estimates toprove global existence of strong solutions of (1.9) in dimensions up to 3, with a classical solutionin 1-dimension. This analysis assumes constant scalar mobility and places restrictions on thetype of well function f used, but includes the prototypical double-well function f ( c ) := ( c − . Addressing problems of integrability, Elliott and Luckhaus [27] proved existence of solutions to(1.9) with the thermodynamically relevant regular solution free energy (1.3). For the analysis ofsolutions to the Cahn-Hilliard equation in the case of degenerate mobility, we direct the readerto [22] and [26].In the results of a workshop in 1990 not published till much later, Fife [31] showed that theCahn-Hilliard equation (1.9) is, in fact, the gradient flow of the energy functional I in the dual2opology of H (Ω) ∩ { ξ : R Ω ξ dx = 0 } , thereby providing a fundamental variational perspective.Making use of such gradient flow structures, Garcke proved existence of a unique, weak solutionto the Cahn-Hilliard equation with elasticity [35]. However, this result was limited in that thepotential f could not be logarithmic. This restriction was lifted in [36], where Garcke treatedweak existence and uniqueness with the inclusion of elasticity and f given by (1.3).A topic of recent interest has been that of the Cahn-Hilliard equation equipped with dynamicboundary conditions, for example ∂ ν c = − ∂ t c + κ ∆ Γ c − g ′ ( c ) on Γ × (0 , T )where ∆ Γ is the Laplace-Beltrami operator and g is a surface potential (see [37] and referencestherein). We also note that there is a variety of work on sharp interface models for both thestatic and evolutionary problems associated with Cahn-Hilliard type energies (see, e.g., [2], [5],[19], [50], [59]).Recently, in [42], Kraus and Roggensack proved existence for a variant of the CHR model(1.5). They assumed constant scalar mobility M , and allow for anisotropy in the interfacialenergy via a positive definite diagonal tensor K , i.e., in (1.1) ρ k∇ c k is replaced by K ( ∇ c ) ·∇ c . They proposed a generalized gradient structure (see [47]) which allows for the inclusion ofhigher order nonlinear boundary conditions in a gradient flow type framework, and proved weakexistence of a solution for finite time intervals. Their variant of the CHR model also includesdamage effects, but is limited by the inclusion of a viscosity term in the chemical potential,which helps to simplify the mathematical analysis. Explicitly, they define µ in (1.11) by µ := − ρ ∆ c + f ′ ( c ) + C ( ce − e ( u )) : e + ǫ∂ t c, (1.10)for some ǫ >
0. Though not used by Bazant et al. (see, e.g., [9], [58]), Kraus and Roggensacknote that the viscosity term ǫ∂ t c can be viewed as a microforce. Lastly, as proposed by Bazantet al. (see, e.g., [21], [58]) the reaction rate R in (1.6) is exponential in µ. The work of Krausand Roggensack is limited to a truncation of the function R which has polynomial growth, and f is also restricted to having polynomial growth.This paper is directed by a motivation to understand the CHR model in dimension N = 3.We begin by extending the work of Kraus and Roggensack and remove the assumption of aviscosity term ǫ∂ t c in µ ; these results hold in arbitrary dimension. Departing from the variationalperspective, we apply fixed point methods to recover strong solutions of the CHR model indimensions N = 2 and 3 for f and R of polynomial growth. Our arguments culminate byshowing, in dimension N = 2, one can recover strong solutions of the CHR model for short timewith f and R defined by (1.3) and (1.6), respectively. The respective question for existence with N = 3 is a work in progress.In order to state our results, we write down the complete CHR models for which we proveexistence. For T >
0, define Ω T := Ω × (0 , T ) and Σ T := Γ × (0 , T ) , where as before Γ := ∂ Ω.Assuming constant scalar mobility and, without loss of generality, that ρ = 1, the CHR modelwith elasticity is given byCHR modelwith elasticity ∂ t c = ∆ µ in Ω T ,µ = − ∆ c + f ′ ( c ) + C ( ce − e ( u )) : e in Ω T , div( C ( e ( u ) − ce )) = 0 in Ω T ,∂ ν c = 0 on Σ T ,∂ ν µ = R ( c, µ ) on Σ T , C ( e ( u ) − ce ) ν = 0 on Σ T ,c (0) = c in Ω . (1.11)3ikewise, the CHR model without elasticity is given byCHR model ∂ t c = ∆ µ in Ω T ,µ = − ∆ c + f ′ ( c ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν µ = R ( c, µ ) on Σ T ,c (0) = c in Ω . (1.12)We obtain existence of weak, strong, and regular solutions using the following notion of weaksolution (for notation we refer the reader to the preliminaries in Section 2). Definition 1.1.
We say that ( c, u ) is a weak solution of the CHR model with elasticity (1.11)in Ω T , if for some δ > c ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) ∩ C ([0 , T ); L (Ω)) ,∂ t c ∈ L (2 − δ ) ′ (0 , T ; H (Ω) ∗ ) ,u ∈ L (2 − δ ) ′ (0 , T ; ˙ H (Ω; R N )) ,c (0) = c ∈ H (Ω) , and for t -a.e. in (0 , T ) the following equations are satisfied for all ξ ∈ H (Ω) and ψ ∈ H (Ω; R N ) : −h ∂ t c ( t ) , ξ i H (Ω) ∗ ,H (Ω) = Z Ω ∇ µ ( t ) · ∇ ξ dx − Z Γ R ( c ( t ) , µ ( t )) ξ d H N − , Z Ω C ( e ( u ( t )) − c ( t ) e ) : e ( ψ ) dx = 0 , (1.13) where for t -a.e., µ ( t ) ∈ H (Ω) ⊂ L (Ω) is defined by duality as ( µ ( t ) , ξ ) L (Ω) := Z Ω ( ∇ c ( t ) · ∇ ξ + f ′ ( c ( t )) ξ + C ( c ( t ) e − e ( u ( t ))) : e ξ ) dx, (1.14) which holds for all ξ ∈ H (Ω) . Remark 1.2.
In the above definition of a weak solution, µ is defined by its action on ξ ∈ H (Ω) versus directly setting µ := − ∆ c + f ′ ( c ) + C ( ce − e ( u )) : e . This is to guarantee that theboundary condition ∂ ν c = 0 is satisfied. Remark 1.3.
One could alternatively define the weak solution as the triple ( c, u, µ ) , where µ ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) , such that (1.14) is satisfied, i.e., (1.14) is no longer a definitionbut instead a relation between c , u , and µ . In Kraus and Roggensack [42], the introduction ofviscosity renders this approach necessary. We now state the existence of weak and strong solution under hypotheses that may be foundin Subsection 2.2
Theorem 1.4.
Let Ω ⊂ R N be a bounded, open domain with C boundary and T > . Supposethat f and R satisfy assumptions (2.1), (2.3), (2.4), and (2.5). Then for any c ∈ H (Ω) , aweak solution of the CHR model with elasticity (1.11) exists in Ω T . The following results are stated for domains with C ∞ or smooth boundary. This assumptionsimplifies liftings of boundary conditions, and we speculate that the next two results hold fordomains with C boundaries. 4 heorem 1.5. Let Ω ⊂ R N , where N = 2 or , be a bounded, open domain with smoothboundary and T > . Suppose f and R satisfy assumptions (2.8), (2.9), (2.10), and (2.11).Then for any c ∈ H (Ω) such that ∂ ν c = 0 on Γ , there is a strong solution, given by c ∈ L (0 , T ; H (Ω)) ∩ H (0 , T ; L (Ω)) , of the CHR model (1.12) (or (4.1)) in Ω T . Given the Sobolev embedding theorem in dimensions N = 2 or 3, we directly have thefollowing corollary. This allows us to remove a plethora of restrictive hypotheses on f for shorttime strong existence. Corollary 1.6.
Let Ω ⊂ R N , where N = 2 or , be a bounded, open domain with smoothboundary. Suppose f ∈ C ( R ) and R satisfies assumptions (2.10) and (2.11). For any c ∈ H (Ω) such that ∂ ν c = 0 on Γ , there exists T > such that a strong solution of the CHR model(1.12) (or (4.1)) exists in Ω T . Remark 1.7.
We note that for existence of strong solutions, it is a necessity that the initialdata c satisfy ∂ ν c = 0 on Γ . This is because a strong solution c ∈ H , (Ω T ) (see Subsection2.5) belongs to the space of bounded uniformly continuous functions on the interval [0 , T ] withvalues in H (Ω) (see [45]). By definition of a solution of the CHR model (1.11), ∂ ν c = 0 on Σ T ,and by continuity, we have ∂ ν c ( · ,
0) = ∂ ν c on Γ . Lastly, we address the regularity of solutions. In the case of a constant scalar mobility tensor,we prove that there is a solution to the CHR model (1.5) as proposed by Singh, et. al. [58] with R as in (1.6). This result makes sharp use of the growth provided by the Gagliardo-Nirenberginequality (see Theorem 2.5), and therefore critically relies on smallness estimates developed inthe analysis of strong solutions (see Theorem 4.1). Furthermore, reasoning similar to Remark1.7 requires that the initial data c satisfies an additional compatibility condition. Theorem 1.8.
Suppose Ω ⊂ R is an open bounded set with smooth boundary. Let f and R be defined as in (1.3) and (1.6), respectively. There exists λ = λ (Ω , f, R ) > such that if c ∈ H (Ω) such that ∂ ν c = 0 , ∂ ν (∆ c ) = − R ( c , − ∆ c + f ′ ( c )) , ǫ ≤ c (Ω) ≤ − ǫ in Ω for some ǫ > , and k∇ c k L (Ω) ≤ λ, then there are T > and c ∈ L (0 , T ; H (Ω)) ∩ H / (0 , T ; L (Ω)) for which c is strong solution of the CHR model (1.12) in Ω T . This paper is organized as follows: In the preliminary Section 2, we recall some ellipticestimates to be used throughout the paper. We introduce fractional Sobolev spaces and a classof anisotropic Sobolev spaces, which will be employed in Sections 4 and 5. In Section 3, weprove weak existence of a solution to the CHR model with (1.11) and without elasticity (1.12),using the generalized gradient structure proposed by Kraus and Roggensack [42]. Our analysisshows how to remove the assumption of a viscosity term introduced in (1.10). This result holdsin arbitrary dimensions (see Theorem 1.4). In Section 4, restricted to N = 2 and 3 , we argue viafixed point and interpolation methods to prove existence of a strong solution to the CHR model(1.12) (see Theorem 1.5 and Corollary 1.6). Finally, in Section 5, and making a priori estimatesderived in the previous section, we show that in dimension N = 2, for sufficiently small intervalsof time and initial data close to a small energy state, we have a strong solution of the CHRmodel (1.12) for R with exponential growth in µ (see Theorem 1.8).We note that Section 3 and Sections 4 and 5 may be read independently. We also note thatthe appendix, in Section 6, analyzes the regularity of a fourth order PDE via a gradient flow inthe dual of H . Hence, for those unfamiliar with gradient flows or differential inclusions, it maybe of use to read Theorem 6.2 before Section 3.
In the first subsection, we list notation used throughout the paper. In Subsection 2.2, we stateassumptions used in Theorems 1.4 and 1.5. In Subsection 2.3, we highlight some elliptic and5mbedding estimates we will use in the following sections. We further introduce function spacesin Subsections 2.4 and 2.5, which will be critical in Sections 4 and 5; these results will not beneeded in Section 3. Here, we remind the reader of fractional Sobolev spaces in the 1-dimensionalsetting. We derive an extension result for such spaces in Corollary 2.8. We then recall a classof anisotropic Sobolev spaces used ubiquitously by Lions and Magenes [46]. Integrating ourknowledge of the two spaces, we propose a new semi-norm for the anisotropic Sobolev spaces tobe used in the later sections.
We enumerate the variety of notation used throughout the paper.1. Given a Banach space B , we let B ∗ denotes the dual space of B . We denote duality betweenthese spaces by h· , ·i B ∗ , B .2. Given Banach spaces ( B , k · k ) and ( B , k · k ), we denote the continuous embedding of B into B by B ֒ → B , and the compact embedding of B into B by B ֒ → ֒ → B
3. In a Hilbert space H , we denote the inner product by ( · , · ) H . H N is the Hausdorff measure of dimension N. See Evans and Gariepy [30] for more infor-mation.5. Given a domain Ω ⊂ R N specified by context, we define Γ := ∂ Ω . We let ν denote theoutward normal of Γ.6. For Ω ⊂ R N specified by context and T > , we define Ω T := Ω × (0 , T ) . Furthermore, weset Σ T := Γ × (0 , T ) .
7. We interchangably use ∇ c as a row vector and a column vector, to be understood fromcontext (though most often a row vector).8. Pos(N) denotes the set of positive definite matrices acting on R N .
9. Given a Banach space B and a < b ∈ R , with J := ( a, b ), let L p ( a, b ; B ) = L p ( J ; B ) denotethe space of Bocher p − integrable functions on J with values in B . For a good resource onsuch spaces, we refer the reader to [44].10. Given a Banach space B and T >
0, let
BU C (0 , T ; B ) denote the space of boundeduniformly continuous functions with values in B on the closed interval [0 , T ] .
11. The space of k − differentiable continuous functions on Ω ⊂ R N with values in a Banachspace B will be denoted by C k (Ω; B ) . If B = R , we abbreviate it as C k (Ω) . If we additionallyrestrict ourselves to functions such that derivatives up to (and including) the k th order arebounded and the k th order derivatives are H¨older continuous with parameter α , the spaceis denoted by C k,α (Ω; B ) , and we set k g k C k,α := X i ≤ k k∇ i g k ∞ + |∇ k g | C ,α .
12. We say that a set Ω ⊂ R N has C k boundary if, for every x ∈ Γ there is some r >
0, suchthat up to rotation, B ( x, r ) ∩ Ω coincides with the epigraph of a C k ( R N − ) function. Inthe case that k = ∞ , we say that Ω has smooth boundary.13. We let Tr : H k (Ω) → H k − / (Γ) denote the trace operator. See [44] for more information.64. As is standard, we let p ∗ := NpN − p be the critical exponent for the Sobolev embedding indimension N > p.
We further let p be the critical value of q for which the trace operator,Tr, continuously maps W ,p (Ω) into L q (Γ). Note for q < p , the embedding is compact.We specifically note 2 := N − N − if N ≥ , any q > N = 2 , ∞ if N = 1 . We refer the reader to [44].15. We use C to denote a generic constant, which can change from line to line. If dependenceof C on parameter a is emphasized, we will denote this by either C a or C ( a ).16. ˙ H k (Ω; R N ) is the Sobolev space quotiented by skew affine functions. We will make use of a collection of assumptions to prove weak existence. • We assume that the chemical energy density is governed by a function f ∈ C ( R ) suchthat, for some C > f ≥ − C and | f ′′ ( s ) | ≤ C ( | s | ∗ / − + 1) (2.1)for all s ∈ R , where 2 ∗ is the dimension dependent Sobolev exponent, 2 ∗ = NN − if N ≥ N ≤ • For the reaction rate R, we assume that there is G ∈ C ( R ) such that ∂ w G ( s, w ) = R ( s, w ) (2.2)for all s ∈ R and w ∈ R . We suppose that the reaction rate is strictly decreasing in thesecond variable, i.e., there is
C > R ( s, w ) − R ( s, w )) ( w − w ) ≤ − C | w − w | (2.3)for all s ∈ R and w , w ∈ R . Further, for some C, δ > , the growth condition | R ( s, w ) | ≤ C ( | s | − δ − + | w | − δ − + 1) (2.4)holds for all s ∈ R and w ∈ R , where 2 := N − N − if N ≥ is any fixed constantgreater than 2 if N ≤
2. We assume there is a constant
C > | R ( s, ± | ≤ C (2.5)is satisfied for any choice of s ∈ R . Testing (2.3) with w = 1 or − w = 0, and using (2.5) ,we find that | R ( s, | ≤ C for some constant C > . From this bound and (2.3) it follows there is
C > s ∈ R and w ∈ R , − wR ( s, w ) ≥ C | w | − C. (2.6)To derive growth conditions of the function G, we may, without loss of generality, supposethat ∂ s G ( s,
0) = R ( s, . Consequently, the fundamental theorem of calculus, (2.4), andYoung’s inequality imply | G ( s, w ) | ≤ C ( | s | − δ + | w | − δ + 1) (2.7)for all s ∈ R and w ∈ R . emark 2.1. If we assume that R ∈ C ( R ) , then (2.2) is immediately satisfied with G ( s, w ) := Z w R ( s, ρ ) dρ. We note these assumptions are in accordance with those of Kraus and Roggensack (see [42]).To prove strong existence, we will make use of more powerful assumptions: • We assume that the chemical energy density is governed by a function f ∈ C ( R ), suchthat for some C > k f ′′ k ∞ + k f ′′′ k ∞ ≤ C. (2.8)We also make use of the coercivity assumption f ( s ) ≥ δ | s | − /δ, (2.9)which holds for all s ∈ R and some δ > . • For the reaction rate, we assume that R is Lipschitz, i.e., there is C > k∇ R k ∞ ≤ C. (2.10)Furthermore, we introduce the growth condition − wR ( s, w ) ≥ − C (2.11)for all s ∈ R and w ∈ R . Remark 2.2.
In comparison with those used to obtain weak existence, it is clear that (2.8)imposes much greater restrictions on the growth of f ′ . This condition arises because we will needto obtain sufficient regularity of the boundary term R ( c, µ ) = R ( c, − ∆ c + f ′ ( c )) . Furthermore,the regularity assumptions on R are more stringent, but we are relatively lenient on the structureof R (i.e., we do not need monotonicity). Note that (2.11) is a relaxation of the condition (2.6). Remark 2.3.
In view of (2.10), | R ( s, w ) | ≤ | R (0 , | + C ( | s | + | w | ) . In particular, by (2.8), | R ( s, − w + f ′ ( s )) | ≤| R (0 , | + C ( | s | + | w | + | f ′ ( s ) | ) ≤| R (0 , | + C ( | s | + | w | + | f ′ (0) | ) . Furthermore, by the chain rule, (2.8), and (2.10), the map ( s, w ) R ( s, − w + f ′ ( s )) is Lipschitz. The following result on elliptic regularity may be found in [39] (see also [45], [48]).
Theorem 2.4.
Let Ω ⊂ R N be an open, bounded set with C k +2 boundary for k ∈ N . Let g ∈ H k (Ω) with R Ω g dx = 0 . Let v ∈ H (Ω) be a weak solution of ( − ∆ v = g in Ω ,∂ ν v = 0 on Γ , (2.12) with R Ω v dx = 0 . Then there is a constant C > , depending only on Ω and N , such that k v k H k +2 (Ω) ≤ C k g k H k (Ω) . (2.13)8e will also make use of the Gagliardo-Nirenberg inequality [53] (sometimes referred to as justthe Nirenberg inequality), which improves upon more the standard Sobolev-Gagliardo-Nirenbergembedding theorem (see, e.g., [44]). Theorem 2.5 (Gagliardo-Nirenberg inequality) . Suppose that Ω ⊂ R N is an open, bounded setwith Lipschitz boundary. Then the following inequality is satisfied for measurable functions v : k∇ j v k L p (Ω) ≤ C k∇ m v k aL r (Ω) k v k − aL q (Ω) + C k v k L q (Ω) , with a ≥ satisfying jm ≤ a ≤ , p = jn + a (cid:18) r − mn (cid:19) + (1 − a ) 1 q . Emulating an argument used in Fonseca et al. [32], we use the previous two results toobtain an interpolation result in the dual of H (Ω) . An inequality of this type also follows frominterpolation theory, but rather than invoking it, we prove this inequality directly in an effortto stay self-contained where possible.
Corollary 2.6.
Let Ω ⊂ R N be an open, bounded set with C boundary. Then g ∈ H (Ω) satisfies the bound k g k L (Ω) ≤ C (cid:16) k g k / H (Ω) k g k / H (Ω) ∗ + k g k H (Ω) ∗ (cid:17) . (2.14) Proof.
First, suppose R Ω g dx = 0 . Let v ∈ H (Ω) be the weak solution of (2.12). Consequentlyfor all ξ ∈ H (Ω) we have Z Ω ∇ v · ∇ ξ dx = Z Ω gξ dx. Taking ξ = v gives k∇ v k L (Ω) = Z Ω gv dx ≤ k g k H (Ω) ∗ k v k H (Ω) ≤ C k g k H (Ω) ∗ k∇ v k L (Ω) , where the last inequality follows from the Poincar´e inequality since R Ω v dx = 0 . It follows that k∇ v k L (Ω) ≤ C k g k H (Ω) ∗ . (2.15)We apply the Gagliardo-Nirenberg inequality (Theorem 2.5), (2.13), and (2.15) to bound k∇ v k L (Ω) ≤ C (cid:16) k∇ v k / H (Ω) k∇ v k / H (Ω) + k∇ v k H (Ω) (cid:17) ≤ C (cid:16) k g k / H (Ω) k g k / H (Ω) ∗ + k g k H (Ω) ∗ (cid:17) . As − ∆ v = g by (2.12), the bound (2.14) follows.If R Ω g dx = 0 , we consider inequality (2.14) for g − R Ω g dx . Noting that (cid:12)(cid:12)(cid:12)(cid:12)Z Ω g dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k H (Ω) ∗ k k H (Ω) = C (Ω) k g k H (Ω) ∗ , the bound (2.14) follows for g by applications of the triangle inequality and subadditivity of thesquare-root. 9 .4 Fractional Sobolev Spaces In this subsection, we develop a sufficient knowledge of fractional Sobolev spaces to introduce anon-standard norm for anisotropic Sobolev spaces in Subsection 2.5 (see (2.24)). As such, ourconsideration is restricted to one-dimensional fractional Sobolev spaces H s (0 , T ) for s ∈ (0 , a, b ) ⊂ R , we define a semi-norm via the difference quotient introduced byGagliardo (see, e.g., [25], [34], [44]) for s ∈ (0 , | u | H s ( a,b ) := (cid:18)Z ba Z ba | u ( x ) − u ( y ) | | x − y | s dy dx (cid:19) / . (2.16)We then define the norm k u k H s ( a,b ) := k u k L ( a,b ) + | u | H s ( a,b ) . (2.17)The space generated by the closure in this norm of H ( a, b ) in L ( a, b ) is the fractional Sobolevspace of order s denoted by H s ( a, b ) . We prove a result for one-dimensional fractional Sobolevspaces.
Proposition 2.7.
Suppose u ∈ H s (0 , T ) and ψ ∈ C ∞ [0 , T ] . Then ψu ∈ H s (0 , T ) , and for any ǫ ∈ (0 , T ) , it satisfies the bound k ψu k H s (0 ,T ) ≤ C ((1 + ǫ − s ) k ψ k ∞ + ǫ − s k∇ ψ k ∞ ) k u k L (0 ,T ) + k ψ k ∞ | u | H s (0 ,T ) . Proof.
As the control of the L norm of ψu is straightforward, we estimate the seminorm | ψu | H s (0 ,T ) as follows. Z T Z T | ψ ( x ) u ( x ) − ψ ( t ) u ( t ) | | x − t | s dx dt ≤ Z T Z T | ψ ( x ) | | u ( x ) − u ( t ) | | x − t | s dx dt + 2 Z Z {| x − t |≤ ǫ } | u ( t ) | | ψ ( x ) − ψ ( t ) | | x − t | s dx dt + 2 Z Z {| x − t | >ǫ } | u ( t ) | | ψ ( x ) − ψ ( t ) | | x − t | s dx dt. To bound the terms of the right-hand side, we immediately have Z T Z T | ψ ( x ) | | u ( x ) − u ( t ) | | x − t | s dx dt ≤ k ψ k ∞ | u | H s (0 ,T ) . For the second term, we use the mean value theorem and Fubini’s theorem to find
Z Z {| x − t |≤ ǫ } | u ( t ) | | ψ ( x ) − ψ ( t ) | | x − t | s dx dt ≤k∇ ψ k ∞ Z T | u ( t ) | (cid:18)Z { x ∈ (0 ,T ): | x − t |≤ ǫ } | x − t | − s dx (cid:19) dt ≤k∇ ψ k ∞ Z T | u ( t ) | dt Z ǫ − ǫ | ζ | − s dζ = C k∇ ψ k ∞ k u k L (0 ,T ) ǫ − s . The third term is directly estimated as
Z Z {| x − t | >ǫ } | u ( t ) | | ψ ( x ) − ψ ( t ) | | x − t | s dx dt ≤ C k ψ k ∞ ǫ − s k u k L (0 ,T ) . Combining the above inequalities, we conclude the lemma.10s a result of the above lemma and a reflection argument, we obtain the following extensionresult.
Corollary 2.8.
Let < T ≤ T . Suppose that u ∈ H s (0 , T ) . There exists an extension of u , ˜ u ∈ H s (0 , T ) , such that the following bound is satisfied: k ˜ u k H s (0 ,T ) ≤ C T (cid:16) (1 + T − s ) k u k L (0 ,T ) + | u | H s (0 ,T ) (cid:17) , (2.18) with constant C T independent of T . We prove an estimate that is helpful for comparing the semi-norm of a fractional Sobolevspace to that of the standard Sobolev space.
Proposition 2.9.
Given u ∈ H (0 , T ) and s ∈ (0 , , we have the following estimate | u | H s (0 ,T ) ≤ s p − s ) T − s k ∂ t u k L (0 ,T ) . Proof.
By definition of the semi-norm and the fundamental theorem of calculus, | u | H s (0 ,T ) = Z T Z T | u ( x ) − u ( y ) | | x − y | s dy dx = Z T (cid:18)(cid:18)Z x + Z Tx (cid:19) | R yx ∂ t u dσ | | x − y | s dy (cid:19) dx. (2.19)Fixing x ∈ (0 , T ) , we bound the x variable’s integrand using the change of variables ¯ y = y − x and ¯ σ = σ − x , and we use Hardy’s inequality (see [44]). To be precise, Z Tx | R yx ∂ t u dσ | | x − y | s dy = Z T − x | R x +¯ yx ∂ t u dσ | | ¯ y | s d ¯ y = Z T − x | R ¯ y ∂ t u ( x + ¯ σ ) d ¯ σ | | ¯ y | s d ¯ y ≤ (1 /s ) Z T − x ¯ y − s | ∂ t u ( x + ¯ y ) | d ¯ y = (1 /s ) Z Tx | x − y | − s | ∂ t u ( y ) | dy. By the same argument for the other integral, we obtain | u | H s (0 ,T ) ≤ (1 /s ) Z T Z T | x − y | − s | ∂ t u ( y ) | dy dx =(1 /s ) Z T (cid:18)Z T | x − y | − s dx (cid:19) | ∂ t u ( y ) | dy ≤ s p − s ) ! T − s k ∂ t u k L (0 ,T ) , concluding the result.We lastly make note of a simple lemma, which shows how the semi-norm changes for arescaled domain. 11 emma 2.10. Let < T , s ∈ (0 , , and u ∈ H s (0 , T ) . Define u T ( x ) := u ( T x ) for x ∈ (0 , .Then it holds that | u | H s (0 ,T ) = T − s | u T | H s (0 , . Proof.
We compute | u | H s (0 ,T ) = Z T Z T | u ( x ) − u ( y ) | | x − y | s dx dy = Z Z | u ( T x ) − u ( T y ) | | T x − T y | s T dx T dy = T − s | u T | H s (0 , . Anisotropic Sobolev spaces naturally arise in the study of PDEs. Let Ω ⊂ R N be a smoothdomain. We recall the notation Ω T := Ω × (0 , T ) , Σ T = ∂ Ω × (0 , T ) , and for r, s ≥ , we define H r,s (Ω T ) := L (0 , T ; H r (Ω)) ∩ H s (0 , T ; L (Ω)) , (2.20)where H r (Ω) and H s (0 , T ; L (Ω)) are defined via interpolation of Sobolev spaces of integer order(see [46]). Note that H (Ω) = L (Ω). Likewise, we may define the anisotropic Sobolev spacewith domain Σ T . It is standard to endow H r,s (Ω T ) with the norm arising from interpolation(denoted by “ , I ”) given by k u k H r,s (Ω T ) ,I := (cid:18)Z T k u ( · , t ) k H r (Ω) ,I dt + k u k H s (0 ,T ; L (Ω)) ,I (cid:19) / . (2.21)As an aside, we recall how interpolation and the interpolation norm are defined. Remark 2.11.
Generically, suppose that X and Y are separable Hilbert spaces, with X denselyembedded into Y. There is a positive, self adjoint, and unbounded operator Λ on Y such that X = dom(Λ) , and the norm on X is equivalent to the graph norm of Λ :1 C k x k X ≤ ( k x k Y + k Λ x k Y ) / ≤ C k x k X . For the construction of such an operator, we refer the reader to [24], [45], [55]. Using spectraltheory for unbounded operators (see [24] and references therein), we may consider fractionalpowers of the operator Λ . Then the interpolation space [ X, Y ] θ for θ ∈ [0 , is defined by [ X, Y ] θ := dom(Λ − θ ) , (2.22) with norm k · k [ X,Y ] θ = ( k · k Y + k Λ − θ · k Y ) / . (2.23) In the context of Sobolev spaces (see Proposition 2.13), for example, we have k · k H / (Ω) ,I := k · k [ H (Ω) ,H (Ω)] / . We note that the interpolation space defined by (2.22) is norm equivalent to that defined bythe K-method (see, e.g., [44], [45]); however, the norm (2.23) is more directly related to normsarising from the Fourier transform.
12e may also define the anisotropic Sobolev space (2.20) on the larger cylindrical domainΩ × R . For u within such a space, we may consider the Fourier transform of u in the variable t given by the Bochner integral ˆ u ( ξ ) := 1(2 π ) / Z R e − iξt u ( t ) dt, and define the Fourier norm (denoted by “ , F”) on the space H s ( R ; L (Ω)) (see also [45]) by k u k H s ( R ; L (Ω)) , F := (cid:18)Z R (1 + | ξ | ) s k ˆ u ( ξ ) k L (Ω) dξ (cid:19) / . We will need precise results about the extension properties of the function spaces in (2.20).Consequently, norms akin to (2.19) will prove to be more useful. Let k · k H r (Ω) denote anystandard norm choice for H r (Ω) . Lemma 2.12.
Let Ω ⊂ R N be an open, bounded set with smooth boundary and r ≥ . For s ∈ (0 , , we recall (2.17) to define the norm k u k H r,s (Ω T ) := (cid:18)Z T k u ( · , t ) k H r (Ω) dt + Z Ω k u ( x, · ) k H s (0 ,T ) dx (cid:19) / . (2.24) The norm (2.24) is equivalent to the norm defined by (2.21). The same is true on domains Σ T . Proof.
By classical results, we have that k · k H r (Ω) ,I is equivalent to k · k H r (Ω) (see [44] for a proofin the case Ω = R N ; the following argument proves the result for extension domains Ω). Thus,it suffices to take r = 0 and prove that the norm k · k H ,s (Ω T ) is equivalent to k · k H ,s (Ω T ) ,I By Corollary 2.8, there is an extension operator T , defined via reflection and truncationin the variable t (for each fixed point x ∈ Ω), such that T : H , (Ω T ) → H , (Ω × R ) and T : H , (Ω T ) → H , (Ω × R ) are linear and bounded. By interpolation (see Theorem 5.1 ofChapter 1 in [45]), it follows that T : H ,s (Ω T ) → H ,s (Ω × R ) is linear and bounded in thetopology of the interpolation norm (2.21). By a direct computation in the spirit of Corollary 2.8,we have that T is also continuous in the topology defined by the norm (2.24). Consequently,using the equivalence of the Gagliardo norm and Fourier norm on R (see [44]) and Fubini’stheorem, we have k u k H ,s (Ω T ) ≤ C kT u k H ,s (Ω × R ) ≤ C kT u k H s ( R ; L (Ω)) , F ≤ C kT u k H s ( R ; L (Ω)) ,I ≤ C k u k H s (0 ,T ; L (Ω)) ,I (2.25)(see subsection 7.1 of Chapter 1 in [45] for equivalence of Fourier and interpolation norm).To obtain the reverse inequality to prove equivalence of the norms, we may essentially reversethe sequence of inequalities in (2.25). In the same way that T was shown to be continuous, wemay show that the restriction operator π : u u | Ω T mapping from H ,s (Ω × R ) to H ,s (Ω T ) iscontinuous in the interpolation norm. Consequently, k u k H s (0 ,T ; L (Ω)) ,I = k π ( T u ) k H s (0 ,T ; L (Ω)) ,I ≤ C kT u k H s ( R ; L (Ω)) ,I ≤ C kT u k H s ( R ; L (Ω)) , F ≤ C kT u k H ,s (Ω × R ) ≤ C k u k H ,s (Ω T ) , where we have once again used the equivalence of the Gagliardo and Fourier norms.We will make use of some interpolation theorems, which provide regularity of certain quan-tities. The following result is Proposition 2.1 of Chapter 4 in [46].13 roposition 2.13. Let Ω be an open, bounded set with smooth boundary. For r, s ≥ and θ ∈ (0 , , we have [ H r,s (Ω T ) , H , (Ω T )] θ = H (1 − θ ) r, (1 − θ ) s (Ω T ) . The same is true on domains Σ T . Theorem 2.14.
Let Ω be an open, bounded set with smooth boundary. Let u ∈ H r,s (Ω T ) with r > / , s ≥ . If j is an integer such that ≤ j < r − / , we may define the j th normalderivative ∂ jν u ∈ H µ j ,λ j (Σ T ) , where µ j r = λ j s = r − j − / r . (2.26) Furthermore, the map u ( ∂ jν u ) { ≤ j Proposition 2.15. Let Ω ⊂ R N be a bounded smooth domain. Suppose that u ∈ H k,k/ (Ω T ) , k ∈ N . Then, ∇ u ∈ H k − , ( k − / (Ω T ) , with the map u 7→ ∇ u continuous in the respectivetopologies.Proof. Let T : H k (Ω) → H k ( R N ) be a linear extension operator (defined via reflection and apartition of unity as in Theorem 13.4 and Remark 13.5 of [44]), such that there is r > T ( ξ ) ⊂ B (0 , r ) for all ξ ∈ H k (Ω) . We extend u as ˜ u ( x, t ) := T ( u ( · , x ))( x ) . Since T ( u ( · , t ))is defined by reflecting u ( · , t ) near ∂ Ω and using a partition of unity (see, e.g., Theorem 13.17 in[44] or Theorem 5.4.1 [29]), the regularity of ˜ u in time is preserved, and so ˜ u ∈ H k,k/ ( R N × (0 , T ))with k ˜ u k H k,k/ ( R N × (0 ,T )) ≤ C k u k H k,k/ (Ω T ) . (2.27)Write x = ( x ′ , x N ) ∈ R N − × R . By identifying ξ ∈ H k,k/ ( R N × (0 , T )) with the function x N ξ (( · , x N ) , · ) , as noted in [46], we may decompose the anisotropic Sobolev space as H k,k/ ( R N × (0 , T )) = H k (cid:0) R ; L ( R N − × (0 , T )) (cid:1) ∩ L (cid:0) R ; H k,k/ ( R N − × (0 , T )) (cid:1) . Consequently, we apply an intermediate derivative theorem (Theorem 2.3 of Chapter 1 in [45])and Proposition 2.13 to conclude that ˜ u maps continuously to ∇ ˜ u ∈ H k − ( R ; L ( R N − × (0 , T ))) ∩ L ( R ; H k − , ( k − / ( R N − × (0 , T ))) . Using continuity of the restriction operator, bound (2.27) concludes the lemma. In this section, we prove existence of weak solutions to the CHR model with and without elastic-ity. In an effort to illuminate the essential techniques, we first focus on the CHR model withoutelasticity. We use a generalized gradient structure first introduced by Kraus and Roggensack toprove existence of solutions to the viscous CHR model in [42]. The gradient structure providesa generalization of the H dual gradient flow proposed by Fife [31]. Following De Giorgi’s min-imizing movements method, we define an implicit scheme to construct approximate solutions ofthe CHR model. As will be seen, letting R = 0 , we can recover the standard implicit schemeused to prove existence of solutions for the Cahn-Hilliard equation (1.9). From here, we derive14 variety of energy estimates allowing us to pass the approximate solutions to the limit, therebyrecovering a weak solution of the CHR model. In Subsection 3.5, elasticity is incorporated bymaking use of regularity theorems for elliptic systems, thereby proving Theorem 1.4.We define our notion of weak solution for the CHR model (1.12) without elasticity. Definition. We say that c is a weak solution of the CHR model (1.12) on Ω T if for some δ > c ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) ∩ C ([0 , T ); L (Ω)) ,∂ t c ∈ L (2 − δ ) ′ (0 , T ; H (Ω) ∗ ) ,c (0) = c ∈ H (Ω) , and for t -a.e. in (0 , T ) the following equation is satisfied for all ξ ∈ H (Ω) : −h ∂ t c ( t ) , ξ i H (Ω) ∗ ,H (Ω) = Z Ω ∇ µ ( t ) · ∇ ξ dx − Z Γ R ( c ( t ) , µ ( t )) ξ d H N − , (3.1) where for t -a.e. µ ( t ) ∈ H (Ω) ⊂ L (Ω) is defined via duality for all ξ ∈ H (Ω) by Z Ω µ ( t ) ξ dx := Z Ω ( ∇ c ( t ) · ∇ ξ + f ′ ( c ( t )) ξ ) dx. Subsections 3.1 to 3.4 are devoted to the proof of the following theorem. Theorem 3.1. Let Ω ⊂ R N be a bounded, open domain with C boundary and T > . Suppose f and R satisfy assumptions (2.1), (2.3), (2.4), and (2.5). Then a weak solution of the CHRmodel (1.12) exists in Ω T . Much of the exposition in this subsection follows the work of Kraus and Roggensack. For moredetails we refer the reader to [42]. We do however highlight a new H dual bound in Lemma3.3, which will be essential to apply a compactness argument in the spirit of Aubin-Lions-Simon[57] in Subsection 3.4.We introduce functionals to define the gradient structure. Let A : L − δ (Γ) × L (Ω) → R ∪ {∞} , A ( c, v ) := ( k∇ v k L (Ω) − R Γ G ( c, v ) d H N − if v ∈ H (Ω) , ∞ otherwise. (3.2)This functional is proper, lower semi-continuous, and convex in the second input by assumptions(2.3), (2.2), and (2.7). Furthermore, define B : L − δ (Γ) × H (Ω) → H (Ω) ∗ , hB ( c, v ) , ξ i H (Ω) ∗ ,H (Ω) := Z Ω ∇ v · ∇ ξ dx − Z Γ R ( c, v ) ξ d H N − . (3.3)Under the monotonicity assumption (2.3) on R , we can show for fixed c ∈ L − δ (Γ) that B c ( · ) := B ( c, · ) is strictly monotone, bounded, and coercive. Consequently, we may define (seeLemma 1 in [42]) the bounded and continuous operator B : L − δ (Γ) × H (Ω) ∗ → H (Ω) , B ( c, v ∗ ) := B − c ( v ∗ ) . (3.4)15etting A c ( · ) := A ( c, · ), using (2.2) and (2.4), one can show that v ∗ ∈ ∂ A c ( v ) ⇐⇒ ( v ∗ , ξ ) L (Ω) = hB ( c, v ) , ξ i H (Ω) ∗ ,H (Ω) for all ξ ∈ H (Ω) . Hence, the gradient structure is given by the inclusion − ∂ t c ∈ ∂ A c ( µ ) , which encapsulates equations ∂ t c = ∆ µ and ∂ ν µ = R ( c, µ ) of (1.12). As this is a differentialinclusion, there is a natural dual formulation. Let A ∗ be the Legendre-Fenchel transform of A with respect to the second input, that is, A ∗ : L − δ (Γ) × L (Ω) → R ∪ {∞}A ∗ ( c, v ∗ ) := ( A c ) ∗ ( v ∗ ) = sup v ∈ L (Ω) { ( v ∗ , v ) L (Ω) − A c ( v ) } (3.5)As Kraus and Roggensack detail, ∂ A ∗ c ( v ∗ ) = {B ( c, v ∗ ) } , (3.6)which provides a way to express µ in terms of c and ∂ t c via convex duality. Explicitly, µ = B ( c, − ∂ t c ) . (3.7)We note a technical result, which will be helpful in obtaining energetic bounds. Lemma 3.2 ([42], Lemma 2) . Under the hypotheses of Theorem 3.1, for any c ∈ L − δ (Γ) and v ∗ ∈ L (Ω) , we have the following bound: N (cid:12)(cid:12)(cid:12) Z Ω v ∗ dx (cid:12)(cid:12)(cid:12) ≤ A ∗ c ( v ∗ ) + A c ( B ( c, C for some constant C > depending only on G , R , and Ω . Finally, we conclude this section with the H dual bound. Lemma 3.3. Let Ω ⊂ R N be a bounded, open set with Lipschitz boundary. Assume hypotheses(2.1) to (2.5) hold with A ∗ defined as in (3.5). Suppose that k c k L − δ (Γ) ≤ α. Then there exists C α > such that A ∗ c ( v ∗ ) ≥ C α k v ∗ k (2 − δ ) ′ H (Ω) ∗ − C α . (3.8) Proof. Under the growth assumption (2.7) on G and the definition A in (3.2), for ξ ∈ H (Ω) , we have A c ( ξ ) ≤ C ( k ξ k H (Ω) + k ξ k − δL − δ (Γ) + k c k − δL − δ (Γ) + 1) ≤ C α ( k ξ k − δH (Ω) + 1)where in the second inequality we have used the trace inequality k ξ k L − δ (Γ) ≤ C k ξ k H (Ω) . As A c = ∞ on L (Ω) \ H (Ω) , we compute by definition of the conjugate A ∗ c ( v ∗ ) = sup ξ ∈ H (Ω) n ( v ∗ , ξ ) L (Ω) − A c ( ξ ) o ≥ sup ξ ∈ H (Ω) n ( v ∗ , ξ ) L (Ω) − C α k ξ k − δH (Ω) o − C α . (3.9)16ince 2 − δ > , there is a unique maximizer ξ to the latter supremum, and computing theGateaux derivative, it must satisfy the following relation for all ξ ∈ H (Ω) :( v ∗ , ξ ) L (Ω) = (2 − δ ) C α k ξ k − δ − H (Ω) ( ξ , ξ ) H (Ω) . Furthermore as the right hand side is maximized over the unit ball at ¯ ξ = ξ / k ξ k H (Ω) , thisimplies 1 k ξ k H (Ω) ( v ∗ , ξ ) L (Ω) = k v ∗ k H (Ω) ∗ = (2 − δ ) C α k ξ k − δ − H (Ω) . Consequently, evaluating the supremum in (3.9) at its maximizer, we find A ∗ c ( v ∗ ) ≥ ( v ∗ , ξ ) L (Ω) − C α k ξ k − δH (Ω) − C α ≥ (2 − δ − C α k ξ k − δH (Ω) − C α ≥ (2 − δ − C α ((2 − δ ) C α ) (2 − δ ) ′ k v ∗ k (2 − δ ) ′ H (Ω) ∗ − C α . Up to redefinition of C α , this completes the lemma. We fix T > . For τ = τ n := T /n > 0, with n ∈ N , we define the following iterative scheme. Let c τ = c . Define c iτ = argmin c ∈ L (Ω) (cid:26) I [ c ] + τ A ∗ c i − τ (cid:18) − c − c i − τ τ (cid:19)(cid:27) , (3.10)where I and A ∗ are defined in (1.7) and (3.5), respectively. Motivated by (3.7), we define thediscretized chemical potential by µ iτ := B (cid:18) c i − τ , − c iτ − c i − τ τ (cid:19) . (3.11)To gain an intuition for the minimization scheme, we heuristically consider the case of R = 0(note however this choice of R does not satisfy the strict monotonicity assumption (2.3)). A directcomputation (see the proof of Lemma 3.3) verifies that A ∗ ( v ∗ ) = k v ∗ k H (Ω) ∗ . Consequently, theabove minimization scheme reduces to c iτ = argmin c ∈ L (Ω) n I [ c ] + 12 τ k c − c i − τ k H (Ω) ∗ o . Furthermore, by definition B ( − ∂ t c ) = (∆) − ( − ∂ t c ) , the inverse Laplacian associated with homo-geneous Neumann boundary conditions. Consequently, we obtain the standard implicit schemefor the Cahn-Hilliard equation (1.9) minus fixing the total mass of c . Lemma 3.4. Assume Ω ⊂ R N is a bounded, open set with Lipschitz boundary and hypotheses(2.1) to (2.5) hold. A minimizer, c iτ ∈ H (Ω) , of the iterative scheme (3.10) exists.Proof. The functional minimized is lower semi-continuous under weak convergence of c in H (Ω);thus it remains to prove coercivity. This follows directly from the bound f ≥ − C (see (2.1))and applying Lemma 3.2:12 k∇ c k + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω c dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ I [ c ] + C + τ (cid:12)(cid:12)(cid:12)(cid:12)Z Ω c − c i − τ τ dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω c i − τ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ I [ c ] + C + τ (cid:18) A ∗ c i − τ (cid:18) − c − c i − τ τ (cid:19) + A c i − τ ( B ( c i − τ , (cid:19) + (cid:12)(cid:12)(cid:12)(cid:12)Z Ω c i − τ dx (cid:12)(cid:12)(cid:12)(cid:12) = I [ c ] + τ A ∗ c i − τ (cid:18) − c − c i − τ τ (cid:19) + C ( c i − τ ) . { c iτ } i to define interpolated and piecewise continuousfunctions as follows. We define c τ and c − τ to be the left and right continuous step functions,respectively: c τ ( t ) := ( c τ if t = 0 ,c i +1 τ if t ∈ ( iτ, ( i + 1) τ ] , i = 0 , . . . , n − , (3.12) c − τ ( t ) := c iτ if t ∈ [ iτ, ( i + 1) τ ) , i = 0 , . . . , n − . (3.13)Likewise we define µ τ to be the left continuous step function. Define ˆ c τ to be the piecewise linearinterpolation of the sequence:ˆ c τ ( t ) := ( i + 1) τ − tτ c iτ + t − iττ c i +1 τ if t ∈ [ iτ, ( i + 1) τ ) , i = 0 , . . . , n − . (3.14) Lemma 3.5. Assume Ω ⊂ R N is a bounded, open set with Lipschitz boundary and hypotheses(2.1) to (2.5) hold. The functions c τ , c − τ , and ˆ c τ satisfy the “discrete” Euler-Lagrange equations − ( ∂ t ˆ c τ ( t ) , ξ ) L (Ω) = Z Ω ∇ µ τ ( t ) · ∇ ξ dx − Z Γ R ( c − τ ( t ) , µ τ ( t )) ξd H N − , (3.15)( µ τ ( t ) , ξ ) L (Ω) = Z Ω ( ∇ c τ ( t ) · ∇ ξ + f ′ ( c τ ( t )) ξ ) dx (3.16) for all ξ ∈ H (Ω) and t a.e. in [0 , T ) , and the energy estimate I [ c τ ( t )] + Z t τ ( t )0 (cid:16) A ∗ c − τ ( − ∂ t ˆ c τ ) + A c − τ ( B ( c − τ , (cid:17) ds ≤ I [ c ] , (3.17) where t τ ( t ) := min { kτ : t ≤ kτ } . Proof. Note I [ c iτ ] + τ A ∗ c i − τ (cid:18) − c iτ − c i − τ τ (cid:19) + τ A c i − τ ( B ( c i − τ , ≤ I [ c i − τ ] + τ A ∗ c i − τ (cid:18) − c i − τ − c i − τ τ (cid:19) + τ A c i − τ ( B ( c i − τ , I [ c i − τ ] + τ (cid:16) A ∗ c i − τ (0) + A c i − τ ( B ( c i − τ , (cid:17) = I [ c i − τ ] + τ (0 , B ( c i − τ , L = I [ c i − τ ] , where we have used the fact c iτ is a minimizer (see (3.10)) and Fenchel’s (in)equality [56]. Moving I [ c i − τ ] to the lefthand side, summing up over the inequalities for i = 1 , . . . , n , and using tele-scoping sums, we conclude the energetic bound (3.17). To obtain the Euler-Lagrange equations(3.15) and (3.16), we compute the subdifferential of the minimized equation in H (Ω) ∗ via (3.6):0 ∈ ∂ | c iτ (cid:18) I [ c ] + τ A ∗ c i − τ (cid:18) − c − c i − τ τ (cid:19)(cid:19) ⇐⇒ ∈ ∂I [ c iτ ] − ∂ A ∗ c i − τ (cid:18) − c iτ − c i − τ τ (cid:19) ⇐⇒ ∈ ∂I [ c iτ ] − (cid:26) B (cid:18) c i − τ , − c iτ − c i − τ τ (cid:19)(cid:27) ⇐⇒ ∈ ∂I [ c iτ ] − { µ iτ } Computing ∂I [ c iτ ] via differentiation, we obtain (3.16). Equation (3.15) follows by definition of µ τ in (3.11). 18 .3 Energy estimates We now obtain some energy estimates, which will be useful in passing to the limit in the “dis-crete” Euler-Lagrange equations. Lemma 3.6. Assume Ω ⊂ R N is a bounded, open set with C boundary and hypotheses (2.1) to(2.5) hold. The functions c τ , c − τ , and ˆ c τ defined in (3.12), (3.13), and (3.14) satisfy the followingestimates uniformly in τ : k c τ k L ∞ (0 ,T ; H (Ω)) ≤ C, (3.18) k ∂ t ˆ c τ k L (2 − δ ) ′ (0 ,T ; H (Ω) ∗ ) ≤ C, (3.19) k µ τ k L (2 − δ ) ′ (0 ,T ; H (Ω)) ≤ C, (3.20) k c τ k L (2 − δ ) ′ (0 ,T ; H (Ω)) ≤ C. (3.21) Proof. To see that (3.18) holds, note that ∇ c τ ∈ L ∞ (0 , T ; L (Ω)) by (3.17), and thus by thePoincar´e inequality, we simply need to bound R Ω c τ dx to conclude. Making use of Lemma 3.2and (3.17) once again, we find (cid:12)(cid:12)(cid:12) Z Ω c iτ dx (cid:12)(cid:12)(cid:12) ≤ τ i X j =1 (cid:12)(cid:12)(cid:12) Z Ω c jτ − c j − τ τ dx (cid:12)(cid:12)(cid:12) + Z Ω | c | dx ≤ N τ i X j =1 (cid:16) A ∗ c j − τ ( − c jτ − c j − τ τ ) + A c j − τ ( B ( c j − τ , (cid:17) + T C + Z Ω | c | dx ≤ N Z T (cid:16) A ∗ c − τ ( − ∂ t ˆ c τ ) + A c − τ ( B ( c − τ , (cid:17) ds + C ≤ C ( c )Furthermore, (3.18) implies that c τ is in L ∞ (0 , T ; L − δ (Γ)) by continuity of the trace. Thecoercivity of A ∗ c given by Lemma 3.3 along with the energy estimate (3.17) then conclude (3.19).Making use of the bound kB ( c, v ∗ ) k H (Ω) ≤ C ( k v ∗ k H (Ω) ∗ + 1) , which holds for C > c (see pg. 6 in [42]), in conjunction with (3.19), and thedefinition (3.11) of µ τ provides (3.20).Note for a.e. t ∈ (0 , T ), c τ ( t ) satisfies the Neumann problem ( µ τ = − ∆ c τ + f ′ ( c τ ) in Ω ,∂ ν c τ = 0 on Γ . (3.22)We use the growth condition | f ′ ( s ) | ≤ C ( | s | ∗ / + 1) for all s ∈ R obtained by integrating (2.1), the Sobolev-Gagliardo-Nirenberg embedding theorem, (3.18), andTheorem 2.4 applied to (3.22) to conclude that k c τ k H (Ω) ≤ C (cid:16) k µ τ − f ′ ( c τ ) k L (Ω) + k c τ k H (Ω) (cid:17) ≤ C (cid:16) k µ τ k L (Ω) + k c τ k ∗ L ∗ (Ω) (cid:17) + C ≤ C (cid:16) k µ τ k L (Ω) + k c τ k ∗ H (Ω) (cid:17) + C ≤ C k µ τ k L (Ω) + C. k c τ k L (2 − δ ) ′ (0 ,T ; H (Ω)) ≤ C. (3.23)We show that f ′ ( c τ ) ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) by the assumptions on f given in (2.1). Assume N ≥ N = 2 follows similarly. By the chain rule in Sobolev spaces [44] and H¨older’s inequality,we compute k∇ ( f ′ ◦ c τ ) k L (Ω) = Z Ω | f ′′ ( c τ ) | k∇ c τ k dx ≤ C Z Ω | c τ | ∗ − k∇ c τ k dx + C Z Ω k∇ c τ k dx ≤ C k| c τ | ∗ − k L ( N/ ( N − ′ (Ω) k ( k∇ c τ k ) k L N/ ( N − (Ω) + C ≤ C k c τ k H (Ω) + C, (3.24)where in the last inequality, we have used the following two bounds: k|∇ c τ | k N/ ( N − L N/ ( N − (Ω) = Z Ω k∇ c τ k ∗ dx ≤ k c τ k ∗ H (Ω) , by the Sobolev-Gagliardo-Nirenberg embedding; likewise, ( N/ ( N − ′ = N/ , leading to k| c τ | ∗ − k N/ L N/ (Ω) = Z Ω | c τ | ∗ dx ≤ k c τ k ∗ H (Ω) ≤ C. As desired, (3.23) and (3.24) then imply k f ′ ( c τ ) k L (2 − δ ) ′ (0 ,T ; H (Ω)) ≤ C. We once again make use of Theorem 2.4 for the problem (3.22) along with the previous boundand (3.20) to conclude (3.21). We wish to pass to the limit with respect to τ in the “discrete” Euler-Lagrange equations (3.15)and (3.16). To do this, we will look directly at the underlying compactness result used to obtainthe Aubin-Lions-Simon compactness theorem [57]. For h ∈ R , we introduce the translationoperator T h defined by action on a function g with domain (0 , T ) : T h ( g )( x ) := g ( x + h ) , x ∈ ( − h, T − h ) . (3.25)We further defined the auxillary set O h := (0 , T ) ∩ ( − h, T − h ), the interval of common definitionfor the functions T h ( g ) and g. We recall a result. Theorem ([57], Theorem 5) . Suppose ( B i , k · k B i ) , i ∈ { , , } , are Banach spaces such that B ֒ → ֒ → B ֒ → B . Let p ∈ [1 , ∞ ) and U ⊂ L p (0 , T ; B ) be a bounded set such that kT h ( g ) − g k L p ( O h ; B ) → as h → uniformly for g ∈ U . Then U is relatively compact in L p (0 , T ; B ) . We will not make direct use of this result, but the proof of the above result may be appliedto prove the following lemma. 20 emma 3.7. Let Ω ⊂ R N be a bounded, open set with Lipschitz boundary. Consider the triple c ∈ H (Ω) Ψ( c ) := ( c, Tr( ∇ c )) ∈ H (Ω) × [ L − δ (Γ)] N × N π ( c, Tr( ∇ c )) := c ∈ L (Ω) ⊂ H (Ω) ∗ . Let p ∈ [1 , ∞ ) and U ⊂ L p (0 , T ; H (Ω)) be a bounded set such that kT h ( c ) − c k L p ( O h ; H (Ω) ∗ ) → as h → uniformly for c ∈ U . Then Ψ( U ) is relatively compact in L p (0 , T ; H (Ω) × [ L − δ (Γ)] N × N ) . Proof. The proof of this lemma primarily follows as in the proof of Theorem 5 in [57]. We donot repeat the entire proof, but show why it still holds, despite the mappings no longer beingembeddings. We claim that for every ǫ > C ǫ > c ∈ H (Ω) thebound k Ψ( c ) k H (Ω) × [ L − δ (Γ)] N × N ≤ ǫ k c k H (Ω) + C ǫ k c k H (Ω) ∗ holds (see also Lemma 8 of [57]). We prove this by contradiction. Suppose not; thus for each n ∈ N there is ξ n ∈ H (Ω) such that k Ψ( ξ n ) k H (Ω) × [ L − δ (Γ)] N × N > ǫ k ξ n k H (Ω) + n k ξ n k H (Ω) ∗ . (3.26)Normalizing in H (Ω) , we may assume that k ξ n k H (Ω) = 1 . Thus by compactness of the mapΨ , there is ξ ∈ H (Ω) such that, up to a subsequence (not relabeled), Ψ( ξ n ) → Ψ( ξ ) in H (Ω) × [ L − δ (Γ)] N × N . Additionally, as Ψ( ξ n ) is bounded in H (Ω) × [ L − δ (Γ)] N × N , (3.26)implies ξ n → H (Ω) ∗ . As ξ n → π ◦ Ψ( ξ ) in H (Ω) ∗ , we have that ξ = 0 necessarily (this isa key feature we needed satisfied by the mappings). However, (3.26) shows0 = k Ψ( ξ ) k H (Ω) × [ L − δ (Γ)] N × N = lim n →∞ k Ψ( ξ n ) k H (Ω) × [ L − δ (Γ)] N × N ≥ ǫ, a contradiction. The rest of the proof follows as in [57].We now prove the desired convergences. In this proof, we make use of the composition symbol ◦ to differentiate the behavior of a map like Ψ( c ) from the composition f ′ ( c ) . Lemma 3.8. Assume Ω ⊂ R N is a bounded, open set with C boundary and hypotheses (2.1) to(2.5) hold. There is c ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) ∩ W , (2 − δ ) ′ (0 , T ; H (Ω) ∗ ) such that the functions c τ , c − τ , and ˆ c τ defined in (3.12), (3.13), and (3.14) satisfy the following (up to a subsequence of τ approaching not relabeled): ˆ c τ ⇀ c in L (2 − δ ) ′ (0 , T ; H (Ω)) ∩ W , (2 − δ ) ′ (0 , T ; H (Ω) ∗ ) , (3.27)Ψ( c τ ) , Ψ( c − τ ) , Ψ(ˆ c τ ) → Ψ( c ) in L (2 − δ ) ′ (0 , T ; H (Ω) × [ L − δ (Γ)] N × N ) , (3.28) f ′ ◦ c τ → f ′ ◦ c in L (2 − δ ) ′ (0 , T ; H (Ω)) , (3.29) µ τ ⇀ µ in L (2 − δ ) ′ (0 , T ; H (Ω)) , (3.30) where Ψ( c ) := ( c, Tr( ∇ c )) ∈ H (Ω) × [ L − δ (Γ)] N × N for c ∈ H (Ω) . Proof. We first show that kT h ( c τ ) − c τ k L (2 − δ ) ′ ( O h ; H (Ω) ∗ ) → h → τ (3.31)where T h is defined in (3.25) and O h := (0 , T ) ∩ ( − h, T − h ) . We estimate the L (2 − δ ) ′ normdirectly in the case that | h | < τ and | h | ≥ τ . Without loss of generality, we perform the21omputation for h > O h = (0 , T − h ) . Partition (0 , T ) into n intervals J , . . . , J n oflength τ = T /n . Case h < τ : In this case O h = (0 , T − h ) . Making use of (3.12), the fundamental theorem ofcalculus, properties of the Bochner integral (see Chapter 8 of [44]), and H¨older’s inequality, wehave kT h ( c τ ) − c τ k (2 − δ ) ′ L (2 − δ ) ′ ( O h ,H (Ω) ∗ ) = n − X i =1 Z iτiτ − h k c i +1 τ − c iτ k (2 − δ ) ′ H (Ω) ∗ dt = h n − X i =1 (cid:13)(cid:13)(cid:13)(cid:13)Z J i +1 ∂ t ˆ c τ ds (cid:13)(cid:13)(cid:13)(cid:13) (2 − δ ) ′ H (Ω) ∗ ≤ h n − X i =1 | J i +1 | (2 − δ ) ′ − Z J i +1 k ∂ t ˆ c τ k (2 − δ ) ′ H (Ω) ∗ ds ≤ hτ (2 − δ ) ′ − Z T k ∂ t ˆ c τ k (2 − δ ) ′ H (Ω) ∗ dt. (3.32) Case h ≥ τ : We define some auxillary variables to help with the computation. Let [ h ] τ be h modulo τ , and let k ∈ N satisfy kτ = h − [ h ] τ . Again by (3.12), the fundamental theorem ofcalculus, properties of the Bochner integral [44], and H¨older’s inequality, we find kT h ( c τ ) − c τ k (2 − δ ) ′ L (2 − δ ) ′ ( O h ,H (Ω) ∗ ) = n − k X i =1 Z iτ − [ h ] τ ( i − τ k c i + kτ − c iτ k (2 − δ ) ′ H (Ω) ∗ dt ! + n − k − X i =1 (cid:18)Z iτiτ − [ h ] τ k c i + k +1 τ − c iτ k (2 − δ ) ′ H (Ω) ∗ dt (cid:19) ≤ n − k X i =1 ( τ − [ h ] τ ) | ∪ i + kj = i +1 J j +1 | (2 − δ ) ′ − Z ∪ i + kj = i +1 J j +1 k ∂ t ˆ c τ k (2 − δ ) ′ H (Ω) ∗ ds ! + n − k − X i =1 [ h ] τ | ∪ i + kj = i J j +1 | (2 − δ ) ′ − Z ∪ i + kj = i J j +1 k ∂ t ˆ c τ k (2 − δ ) ′ H (Ω) ∗ ds ! ≤ (2 h ) (2 − δ ) ′ − (cid:18)Z T k ∂ t ˆ c τ k (2 − δ ) ′ H (Ω) ∗ dt (cid:19) ( n − k ) τ. (3.33)Making use of (3.19), (3.32), and (3.33), we conclude the proof of (3.31). Similarly (3.31) holdsfor c − τ and ˆ c τ too.Consequently making use of (3.21), we apply Lemma 3.7 to c τ , c − τ , and ˆ c τ . Thus up to asubsequence of τ (not relabeled), there is c ∈ L (2 − δ ) ′ (0 , T ; H (Ω)) ∩ W , (2 − δ ) ′ (0 , T ; H (Ω) ∗ )such that Ψ( c τ ) , Ψ( c − τ ) , Ψ(ˆ c τ ) → Ψ( c ) in L (2 − δ ) ′ (0 , T ; H (Ω) × [ L − δ (Γ)] N × N ).We remark a priori, it is not clear that c τ , c − τ and ˆ c τ converge to the same c. Let us show that c τ and ˆ c τ converge to the same limit. Suppose c τ → c in L (2 − δ ) ′ (0 , T ; L (Ω)) and ˆ c τ → c in L (2 − δ ) ′ (0 , T ; L (Ω)) . Letting Φ : L (Ω) → H (Ω) ∗ be the natural, continuous inclusion given byΦ( ξ )( φ ) := R Ω ξφ dx for all φ ∈ H (Ω). On the interval J i defined above, we have c τ | J i = ˆ c τ ( iτ ).By the fundamental theorem of calculus, (3.19), and H¨older’s inequality, we compute for any t ∈ J i , k Φ( c τ )( t ) − Φ(ˆ c τ )( t ) k H (Ω) ∗ = k Φ(ˆ c τ )( iτ ) − Φ(ˆ c τ )( t ) k H (Ω) ∗ ≤ Z iτt k ∂ t ˆ c τ ( t ) k H (Ω) ∗ dt ≤ C ( iτ − t ) − δ ≤ Cτ − δ . The above estimate holds for any t ∈ (0 , T ), and we concludeΦ( c τ ) − Φ(ˆ c τ ) → L ∞ (0 , T ; H (Ω) ∗ ) as τ → . c τ ) − Φ(ˆ c τ ) → Φ( c ) − Φ( c ) = Φ( c − c ) in L (2 − δ ) ′ (0 , T ; H (Ω) ∗ ) as τ → . Consequently, Φ( c − c ) = 0 , which by the injectivity of Φ , implies c = c as desired.Furthermore, c τ ’s strong convergence allows us to show (3.29). We show convergence of thegradient ∇ ( f ′ ◦ c τ ), with convergence of f ′ ◦ c τ in L (2 − δ ) ′ (0 , T ; L (Ω)) being easier to conclude.To see this, we apply Lebesgue dominated convergence theorem in an iterative fashion. Considera subsequence of τ such that both c τ → c and ∇ c τ → ∇ c pointwise a.e. in Ω × (0 , T ) and for t -a.e., c τ ( t ) → c ( t ) in H (Ω) . By (2.1) and Young’s inequality |∇ ( f ′ ◦ c τ ) | = | f ′′ ( c τ ) | |∇ c τ | ≤ C ( | c τ | ∗ + |∇ c τ | ∗ + |∇ c τ | ) , and so ∇ ( f ′ ◦ c τ )( t ) → ∇ ( f ′ ◦ c )( t ) in L (Ω) for t -a.e. in (0 , T ) by the generalized Lebesguedominated convergence theorem and the Sobolev-Gagliardo-Nirenberg embedding theorem. Welook at the previously derived bound (3.24), which shows us that for t -a.e., we have k∇ ( f ′ ◦ c τ − f ′ ◦ c ) k L (Ω) ≤ C ( k c τ k H (Ω) + k c k H (Ω) + 1) . As the right-hand side of the above bound converges in L (2 − δ ) ′ (0 , T ; R ) and the left-hand sideconverges to 0 for t -a.e., another application of the generalized Lebesgue dominated convergencetheorem concludes.Convergence given in (3.30) follows from µ τ = − ∆ c τ + f ′ ◦ c τ (see (3.16)) in conjunction withthe convergences (3.27) and (3.29)We consider a definition, which will prove useful in the next proof. Definition 3.9. For Ω ⊂ R N , an open, bounded with Lipschitz boundary, we define the inclusionoperator I : L (2 − δ ) ′ (Γ) → H / (Γ) ∗ by hI ( g ) , ξ i H / (Γ) ∗ ,H / (Γ) := Z Γ gξ d H N − . To see that the inclusion makes sense, for any ξ ∈ H / (Γ) there exists ξ ∈ H (Ω) such that Tr( ξ ) = ξ and C k ξ k H (Ω) ≤ k ξ k H / (Γ) ≤ C k ξ k H (Ω) , for constant C > independent of ξ (seeTheorem 18.40 in [44]). Thus for g ∈ L (2 − δ ) ′ (Γ) , we have (cid:12)(cid:12) hI ( g ) , ξ i H / (Γ) ∗ ,H / (Γ) (cid:12)(cid:12) ≤ k g k L (2 − δ ) ′ (Γ) k ξ k L − δ (Γ) ≤ C k g k L (2 − δ ) ′ (Γ) k ξ k H (Ω) ≤ C k g k L (2 − δ ) ′ (Γ) k ξ k H / (Γ) . We now have enough machinery in place to prove that there is a weak solution of the CHRmodel (1.12). Proof of Theorem 3.1. We prove that the function c from Lemma 3.8 is a weak solution of theCHR model in two steps. First, we prove that the Euler-Lagrange equation (3.1) is satisfied.Second, we prove the initial condition is satisfied. Step 1: Integrating equation (3.15) in time for ξ ∈ L − δ (0 , T ; H (Ω)), we have − Z T ( ∂ t ˆ c τ ( t ) , ξ ) L dt = Z T Z Ω ∇ µ τ ( t ) · ∇ ξ dx dt − Z T Z Γ R ( c − τ ( t ) , µ τ ( t )) ξ d H N − dt, (3.34)23hich makes sense by Lemma 3.6 and (2.4). We wish to pass τ → 0. Making use of Lemma3.8, the only term for which this is difficult is the boundary term. Denote R τ := R ( c − τ , µ τ ) fornotational simplicity. Reconfiguring equation (3.34) and applying H¨older’s inequality, we obtain (cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ R τ ξ d H N − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ µ τ k L (2 − δ ) ′ (0 ,T ; L (Ω)) k∇ ξ k L − δ (0 ,T ; L (Ω)) + k ∂ t c τ k L (2 − δ ) ′ (0 ,T ; H (Ω) ∗ ) k ξ k L − δ ((0 ,T ) ,H (Ω)) . Making use of Lemma 3.6, we concludesup ξ ∈ L − δ (0 ,T ; H (Ω)) k ξ k L − δ (0 ,T ; H ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z T Z Γ R τ ξ d H N − dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C. (3.35)We now consider the naturally defined inclusion operator I : L (2 − δ ) ′ (Γ) → H / (Γ) ∗ (see Def-inition 3.9). By the supremum (3.35), we have that I ( R τ ) is bounded in L − δ (0 , T ; H / (Γ)) ∗ = L (2 − δ ) ′ (0 , T ; H / (Γ) ∗ ) uniformly with respect to τ. Furthermore, as H / (Γ) is reflexive, so is L (2 − δ ) ′ (0 , T ; H / (Γ) ∗ ) [33], and by weak compactness, up to a subsequence, I ( R τ ) ⇀ ζ forsome ζ ∈ L (2 − δ ) ′ (0 , T ; H / (Γ) ∗ ) . We claim that for t -a.e. I ( R τ ) → I ( R ( c, µ )) in H / (Γ) ∗ . Looking to the convergence givenby (3.28), we see that ∆ c τ → ∆ c in the space of real-valued functions L (2 − δ ) ′ (Γ × (0 , T )) , andup to a subsequence of τ (not relabeled), we may apply classical results for L p spaces to concludethat ∆ c τ → ∆ c pointwise H N -a.e. in Γ × (0 , T ) . Furthermore, by definition of the convergence in L (2 − δ ) ′ (0 , T ; L − δ (Γ)), we have k ∆ c τ − ∆ c k L − δ (Γ) → L (2 − δ ) ′ (0 , T ). It follows by classicalresults for L p spaces that up to a subsequence of τ (not relabeled) ∆ c τ ( t ) → ∆ c ( t ) in L − δ (Γ)for t -a.e. in (0 , T ) . Repeating this argument using (3.28), (3.29), and continuity of the trace, wemay assume, up to another subsequence of τ , for t -a.e. c − τ ( t ) → c ( t ) and f ′ ( c τ )( t ) → f ′ ( c )( t )in L − δ (Γ) and pointwise a.e. in the domain Γ. Recalling the growth estimate on R given by(2.4) and (3.22), we have | R τ ( t ) | (2 − δ ) ′ ≤ C ( | c − τ ( t ) | − δ − + | µ τ ( t ) | − δ − + 1) (2 − δ ) ′ ≤ C ( | c − τ ( t ) | − δ + | ∆ c τ ( t ) | − δ + | f ′ ( c τ )( t ) | − δ + 1) . As R is a continuous function, we utilize the generalized Lebesgue Dominated Convergencetheorem to conclude R τ ( t ) → R ( c, µ )( t ) in L (2 − δ ) ′ (Γ) for t -a.e. Continuity of I implies for t -a.e. I ( R τ ( t )) → I ( R ( c, µ )( t )) in H / (Γ) ∗ , proving the claim. Applying Mazur’s Lemma [13], thisfurther implies ζ = I ( R ( c, µ )) . Passing τ to the limit in (3.34) and using the variety of convergences derived herein and inLemma 3.8, we obtain − Z T h ∂ t c, ξ i H (Ω) ∗ ,H (Ω) dt = Z T Z Ω ∇ µ · ∇ ξ dx dt − Z T hI ( R ( c, µ )) , ξ i H / (Γ) ∗ ,H / (Γ) dt. By definition of I , this is rewritten as − Z T h ∂ t c, ξ i H (Ω) ∗ ,H (Ω) dt = Z T Z Ω ∇ µ · ∇ ξ dx dt − Z T Z Γ R ( c, µ ) ξ d H N − dt. Considering a dense collection of { ξ k } k ∈ N ⊂ H (Ω), we let ξ = ξ k (constant in time) in the aboveequation. By a standard analysis using Lebesgue points (see also Theorem 6.2), we find theEuler-Lagrange equation (3.1) is satisfied by c for t -a.e.24 tep 2: The initial condition and continuity of c follows directly from bound (3.18), (3.19), andthe Aubin-Simon-Lions Compactness theorem with p = ∞ [57]. Or we may avoid the use ofhigh-level compactness theorems as follows.Note that k ˆ c τ ( t ) − ˆ c τ ( t ) k H (Ω) ∗ ≤ Z t t k ∂ t ˆ c τ ( t ) k H (Ω) ∗ dt ≤ C ( t − t ) / (2 − δ ) by an application of H¨older’s inequality and (3.19). Applying Corollary 2.6, we find k ˆ c τ ( t ) − ˆ c τ ( t ) k L (Ω) ≤ k ˆ c τ ( t ) − ˆ c τ ( t ) k / H (Ω) k ˆ c τ ( t ) − ˆ c τ ( t ) k / H (Ω) ∗ + k ˆ c τ ( t ) − ˆ c τ ( t ) k H (Ω) ∗ By (3.18), k ˆ c τ ( t ) − ˆ c τ ( t ) k H (Ω) ≤ C. Synthesizing these three inequalities, we have k ˆ c τ ( t ) − ˆ c τ ( t ) k L (Ω) ≤ C max { ( t − t ) / (2(2 − δ )) , ( t − t ) / (2 − δ ) } . (3.36)Passing τ → , we see that c satisfies the relation (3.36) for t -a.e. Furthermore, letting t = 0 in(3.36), we have k ˆ c τ ( t ) − c k L (Ω) ≤ C max { ( t ) / (3(2 − δ )) , ( t ) / (2 − δ ) } . Letting τ → 0, we find that c (0) = c as desired. We complete the proof of Theorem 1.4. We highlight where the argument differs from the proofof Theorem 3.1. Proof of Theorem 1.4. We construct an iterative scheme via the minimization( c iτ , u iτ ) = argmin ( c,u ) ∈ L (Ω) n I el [ c, u ] + τ A ∗ c i − τ (cid:18) − c − c i − τ τ (cid:19) o . (3.37)As before we are able to obtain the “discrete” Euler-Lagrange equations for all ξ ∈ H (Ω) and ψ ∈ H (Ω; R N ): − ( ∂ t ˆ c τ ( t ) , ξ ) L (Ω) = Z Ω ∇ µ τ ( t ) · ∇ ξ dx − Z Γ R ( c − τ ( t ) , µ τ ( t )) ξ d H N − ( µ τ ( t ) , ξ ) L (Ω) = Z Ω ( ∇ c τ ( t ) · ∇ ξ + f ′ ( c τ ( t )) ξ + C ( c τ ( t ) e − e ( u τ ( t ))) : e ξ ) dx, Z Ω C ( e ( u τ ( t )) − c τ ( t ) e ) : e ( ψ ) dx. (3.38)The estimate (3.17) continues to hold, with I replaced by I el . In turn, one has the bounds(3.18), (3.19), and (3.20) of Lemma 3.6. To conclude (3.21), we claim that u τ is bounded in L ∞ (0 , T ; ˙ H (Ω; R N )) uniformly with respect to τ , where ˙ H (Ω) is H (Ω) quotiented by skewaffine functions. Note by (3.17), which holds with I el , and Korn’s inequality [54], we have u τ bounded in L ∞ (0 , T ; ˙ H (Ω; R N )). By the last equation of (3.38), we have Z Ω C ( e ( u τ )) : e ( ξ ) dx = Z Ω C ( c τ e ) : e ( ξ ) dx = Z Ω c τ L ( ∇ ξ ) dx, L : R N × N → R is a linear function. Applying integration by parts, we find Z Ω C ( e ( u τ )) : e ( ξ ) dx = Z Ω L ( ∇ c τ ) · ξ dx + Z Γ L ( c τ ν ) · ξ d H N − , where L : R N → R N and L : R N → R N are linear functions. It follows that u τ is a weaksolution of the PDE ( div[ C ( e ( u τ ))] = L ( ∇ c τ ) in Ω , C ( e ( u τ )) · ν = L ( c τ ν ) on Γ . By regularity results for linearized elastostatic problems on C domains [4] (see also [20]), wehave k u τ k ˙ H (Ω; R N ) ≤ C (cid:0) k L ( ∇ c τ ) k L (Ω; R N ) + k L ( c τ ν ) k H / (Γ; R N ) (cid:1) ≤ C k c τ k H (Ω) , (3.39)which proves the claim by (3.18). With this, we may proceed as in Lemma 3.6 to conclude(3.21). With these estimates, we once again obtain the convergences provided by Lemma 3.8.Estimating u τ k − u τ m , for k, m ∈ N , by a bound analogous to (3.39) and using (3.28), we see that u τ is Cauchy in L (2 − δ ) ′ (0 , T ; ˙ H (Ω)) , and hence converges strongly in L (2 − δ ) ′ (0 , T ; ˙ H (Ω)) to u ∈ L ∞ (0 , T ; ˙ H (Ω)). Thus Tr( e ( u τ )) → Tr( e ( u )) in L (2 − δ ) ′ (0 , T ; L − δ (Γ)).From this, we may proceed as in Theorem 3.1 to pass to the limit in the “discrete” Euler-Lagrange Equations (3.38). In this section, we prove strong existence of solutions to the CHR model (1.12) in dimensions N = 2 and 3 . In this section we will depart from the variational perspective developed in Section3 to prove existence of a weak solution. Although it it possible that a bootstrapping argumentapplied to the weak solution already recovered could lead to a strong solution, restrictions onthe possible choices of f and R would still be governed by properties of composition; hence it isnot clear we would obtain existence of strong solutions under more general hypotheses than inTheorem 1.5. Furthermore, we will see that the limitations highlighted by Remark 5.3 requireus to apply more sophisticated methods than directly bootstrapping to obtain higher regularity.Consequently, we find it instrumental to develop these methods in the simpler context of strongsolutions. Lastly, our proof of regularity in Section 5 will require smallness estimates which wewill derive with the aid of function spaces developed in Section 2 (e.g. in Theorem 4.1).We will make extensive use of Schaefer’s fixed point theorem and interpolation theory. Assuch it will be useful to look at the CHR model (1.12) in the equivalent formulation: ∂ t c + ∆ c = ∆ f ′ ( c ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν (∆ c ) = R ( c, ∆ c ) on Σ T ,c (0) = c in Ω , (4.1)where we define R ( s, w ) := − R ( s, − w + f ′ ( s )) ,µ := − ∆ c + f ′ ( c ) . (4.2)In the language of anisotropic Sobolev spaces, we seek a solution of (4.1) belonging to H , (Ω T ) , so we need ∆ f ′ ( c ) ∈ H , (Ω T ) . As is well known, for Lipschitz f the composition map definedby u f ( u ) is linearly bounded from H (Ω) to H (Ω) , i.e. k f ( u ) k H (Ω) ≤ C f ( k u k H (Ω) + 1) . 26t is then natural to hope that the same would hold of composition operators from H (Ω) to H (Ω) , but this is too much to ask. To illuminate this problem, consider∆ f ′ ( c ) = f ′′′ ( c ) k∇ c k + f ′′ ( c )∆ c. Clearly the second term is linearly bounded by the H (Ω) norm of c , but the first term isquadratic, and it will be impossible to avoid this nonlinearity without modification. Hence for α > 0, we can introduce a truncation function ψ α ∈ C ∞ ( R ) such that ψ α ( x ) = x for all x ∈ ( − α, α ), k ψ ′ α k ∞ ≤ 2, and ψ ′ α = 0 on ( − α − , α + 1) C . We define Ψ α ( x ) := ( ψ α ( x ) , . . . , ψ α ( x N )).We then consider the truncated Laplacian :(∆ f ′ ) α ( c ) := f ′′′ ( c ) k Ψ α ( ∇ c ) k + f ′′ ( c ) ψ α (∆ c ) . (4.3)Although the preceeding discussion shows we don’t need to truncate the second term, we do soas it will be useful in Section 5. In addition to analysis of (4.1), we will consider strong solutionsof the truncated CHR model : ∂ t c + ∆ c = (∆ f ′ ) α ( c ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν (∆ c ) = R ( c, ∆ c ) on Σ T ,c (0) = c in Ω , (4.4)where R is still defined as in (4.2), i.e., the “chemical potential” is unmodified on the boundary.The truncated CHR model (4.4) is relevant for two primary reasons. • It allows us to circumnavigate analytical complications arising from composition, whichwill be especially helpful in proving higher regularity. • If ∇ c and ∇ c are continuous, for T sufficiently small and α well chosen, we will recover asolution to (4.1).After proving the claims of existence given in the introduction (e.g. Theorem 1.5), we intro-duce an a priori estimate, which holds for any solution of the the aforementioned PDEs. Theseestimates will be essential to prove existence of a regular solution in Section 5. These esti-mates show that given initial data sufficiently small (in a specific sense), the solution maintainsquantifiably small energy for short times. Theorem 4.1. Suppose Ω ⊂ R N , where N = 2 or , is a bounded, open set with smoothboundary. Further suppose f and R , defined by (4.2), satisfy assumptions (2.8) and (2.10). Let c ∈ H (Ω) with ∂ ν c = 0 on Γ . Then any strong solution c of the truncated CHR model (4.4)satisfies the estimate k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) ≤ C k ∆ c k L (Ω) + η ( c , T ) , (4.5) for some C > with η ( c , T ) tending to as T → . If additionally f and R satisfy (2.9) and(2.11), then any strong solution c of the CHR model (4.1) satisfies estimate (4.5). We prove existence with the use of Schaefer’s fixed point theorem. To control the bulk data asnecessary we look to the analysis of Elliott and Songmu [28], wherein the Gagliardo-Nirenberginequality (see Theorem 2.5) makes an appearance. The Gagliardo-Nirenberg inequality willprovide us with the means to decompose nonlinear terms that arise from repeated differentiationinto two pieces, typically one controlled in L ∞ and the other in L . Many of the methods appliedherein will be applied once again in the slightly more technical proof of Theorem 5.2.27 roof of Theorem 1.5. We give the full proof in the more delicate case that N = 3 and indicatethe changes in the case N = 2 . Step 1: Assume N = . We define the Banach space B := H , / (Ω T ) . Define the operator A : B → B ,v c by the PDE ∂ t c + ∆ c = ∆ f ′ ( v ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν (∆ c ) = R ( v, ∆ v ) on Σ T ,c (0) = c in Ω . (4.6)To prove existence of a strong solution to CHR model (4.1), we claim that A : B → B satisfiesthe hypotheses of Schaefer’s fixed point theorem [29]. These hypotheses are characterized as • Compactness: The functional A : B → B is compact. • Continuity: The functional A : B → B is continuous. • Bounded: The set { c ∈ B : λA [ c ] = c, λ ∈ (0 , } is bounded in the norm of B .Supposing the claim, by Schaefer’s fixed point theorem, we conclude there is c ∈ B such that A [ c ] = c, and by the argument for compactness, c ∈ H , (Ω T ). Therefore c is a strong solutionof the CHR model 4.1. Thus, it only remains to verify the claim. Compactness: Given Theorem 2.14 and Remark 2.3, it follows kR ( v, ∆ v ) k H / , / (Σ T ) ≤ C (Ω , T ) kR ( v, ∆ v ) k H , / (Ω T ) ≤ C ( R , Ω , T )( k v k H , / (Ω T ) + k∇ v k H , / (Ω T ) + 1) ≤ C ( R , Ω , T )( k v k H , / (Ω T ) + 1) , (4.7)where in the third inequality we have made use of Proposition 2.15 with k = 3. We have∆ f ′ ( v ) = f ′′′ ( v ) k∇ v k + f ′′ ( v )∆ v. (4.8)Using the Gagliardo-Nirenberg inequality (see Theorem 2.5) to control the quadratic term of(4.8), we have k∇ v k L (Ω) ≤ C k∇ v k aL (Ω) k∇ v k − aL (Ω) + C k∇ v k L (Ω) , 14 = a (cid:18) − (cid:19) + (1 − a ) 12 = ⇒ a = 3 / . (4.9)As a < / , it follows from (2.8) and (4.8) that k ∆ f ′ ( v ) k L (Ω) ≤ C ( f ) (cid:16) k∇ v k aL r (Ω) k∇ v k − a ) L (Ω) + k∇ v k L (Ω) + k∇ v k L (Ω) (cid:17) ≤ C (cid:16) k∇ v k L r (Ω) k v k − a ) L ∞ (0 ,T ; H (Ω)) + k v k − a )+1 L ∞ (0 ,T ; H (Ω)) + k∇ v k L (Ω) + 1 (cid:17) . We integrate in time and use that B = H , / (Ω T ) ֒ → L ∞ (0 , T ; H (Ω)) (see Theorem 3.1 ofChapter 1 in [45]) to find k ∆ f ′ ( v ) k H , (Ω T ) ≤ C ( T, f ) (cid:16) k v k − a ) L ∞ (0 ,T ; H (Ω)) k∇ v k L (0 ,T ; L (Ω)) + k v k − a )+1 L ∞ (0 ,T ; H (Ω)) + k∇ v k H , (Ω T ) + 1 (cid:17) ≤ C (cid:16) k v k − a )+1 B + 1 (cid:17) , (4.10)28here in the last inequality we used the fact that 2(1 − a ) + 1 = 11 / > . By (4.6), (4.7), (4.10), and Theorem 6.7 (with k = 0), we have k c k H , (Ω T ) ≤ C ( c ) (cid:16) k v k − a )+1 B + 1 (cid:17) . (4.11)As H , (Ω T ) ֒ → ֒ → B by standard interpolation results (see [44]), bound (4.11) proves compact-ness of the operator A . Continuity: This follows from the generalized Lebesgue dominated convergence theorem andthe estimates derived in showing compactness (see the proof of Theorem 5.2 for details in ananalogous case). Boundedness: Suppose that c ∈ B satisfies λA [ c ] = c ; without loss of generality, we assumethat λ = 1 . As c is a fixed point of A , c ∈ H , (Ω T ) by (4.11) and satisfies (4.1). Arguing viamollification, it is straightforward to show that k∇ c k L (Ω) is absolutely continuous in time (fora related perspective see also pg. 330 of [12]). Consequently, the integral I [ c ( t )] (see (1.7)) isabsolutely continuous as a function of time and may therefore be differentiated. Making useof integration by parts and the embedding H , (Ω T ) ֒ → BU C (0 , T ; H (Ω)) [45], for t -a.e. wedifferentiate the gradient term of the integrand as follows: ∂ t (cid:18) Z Ω k∇ c k dx (cid:19) = lim δ → Z Ω k∇ c ( t + δ ) k − k∇ c ( t ) k δ dx = lim δ → Z Ω ( ∇ c ( t + δ ) , ∇ c ( t + δ )) − ( ∇ c ( t ) , ∇ c ( t ))2 δ dx = lim δ → Z Ω ( ∇ c ( t + δ ) − ∇ c ( t ) , ∇ c ( t + δ )) + ( ∇ c ( t ) , ∇ c ( t + δ ) − ∇ c ( t ))2 δ dx = − lim δ → Z Ω (cid:18) ∆ c ( t + δ ) + ∆ c ( t )2 (cid:19) (cid:18) c ( t + δ ) − c ( t ) δ (cid:19) dx = − Z Ω ∆ c ∂ t c dx Then, we compute the derivative ∂ t I [ c ]( t ) = Z Ω (cid:0) f ′ ( c ) ∂ t c − ∆ c ∂ t c (cid:1) dx = Z Ω (cid:0) f ′ ( c )( − ∆ c + ∆ f ′ ( c )) − ∆ c ( − ∆ c + ∆ f ′ ( c )) (cid:1) dx = − Z Ω (cid:0) k∇ f ′ ( c ) k − ∇ f ′ ( c ) · ∇ (∆ c ) + k∇ (∆ c ) k (cid:1) dx − Z ∂ Ω ∂ ν (∆ c ) µ d H ≤ − Z Ω k∇ µ k dx + C ( R ) H ( ∂ Ω) , where we have used (2.11) and (4.6). It follows that I [ c ( t )] ≤ I [ c ] + T C ( R ) H ( ∂ Ω) (4.12)for all t ∈ (0 , T ) . By the coercivity of f in (2.9), the Poincar´e inequality implies k c k L ∞ (0 ,T ; H (Ω)) ≤ C ( T, R, δ )( I [ c ] + 1) = C ( T, R, δ, c ) . We use the above estimate and the first inequality of (4.10) to conclude k ∆ f ′ ( c ) k H , (Ω T ) ≤ C ( T, f, R, δ, c ) k c k H , (Ω T ) + C ( T, f, R, δ, c ) . (4.13)29pplying Theorem 6.7 (with k = 0) to (4.6) as before, we control c in terms of its data: k c k H , (Ω T ) ≤ C ( kR ( c, ∆ c ) k H / , / (Σ T ) + k ∆ f ′ ( c ) k H , (Ω T ) ) . Using (4.7) and (4.13), we have k c k H , (Ω T ) ≤ C ( R, f )( k c k H , / (Ω T ) + 1) + C ( T, f, R, δ, c ) . The interpolation inequality k c k H , / (Ω T ) ≤ C k c k / H , (Ω T ) k c k / H , (Ω T ) (see [45]) and the particularYoung’s inequality a / b / ≤ ǫb + C ( ǫ ) a for ǫ > ǫ sufficiently small k c k H , (Ω T ) ≤ C ( T, f, R, δ, c , ǫ, Ω) . Step 2: Assume N = . The only dimension dependent inequality arising in the above argu-ment was the Gagliardo-Nirenberg inequality. In inequality (4.9), a is now a = 1 / . This is ofcourse sufficient to repeat the above proof. Proof of Corollary 1.6. Given c ∈ H (Ω) , we have c ∈ C ,α (Ω) by the Morrey embeddingtheorem [44]. Choose ˜ f which satisfies hypotheses (2.8) and (2.9) and ˜ f | O = f , where O =( −k c k C ,α − , k c k C ,α + 1) . Applying Theorem 1.5, there is c ∈ H , (Ω ) a solution of the CHRmodel 4.1 with ˜ f . As H , (Ω ) ֒ → BU C (0 , H (Ω)) ֒ → BU C (0 , C ,α (Ω)) [45], for sufficientlysmall T > c | Ω T is a solution of the CHR model (1.12) with f. We emulate the above proof of boundedness keeping estimates of smallness to show that a solutionwith small data and short time stays small in energy. Two challenges occur which make theproof of the following more involved: • We know that k v k L ∞ (0 ,T ; H (Ω)) ≤ C (?) k v k H , (Ω T ) from the trace theory detailed by Lionsand Magenes (see [45], [46]). Necessarily though, C (?) depends on T , blowing up as T → • Returning to the notation of subsection 2.5, there is insufficient literature detailing theconstants by which k · k H s (0 ,T ) ,I is equivalent to k · k H s (0 ,T ) , where the later norm is given bythe integral of the derivative or difference quotients. Existing results of which the authorsare aware address this relation with the use of extensions (see, e.g., [18]).As we will send T → , i.e., shrink the size of our domain, these constants are critical. Tonavigate this problem, we use a variety of estimates developed in Subsection 2.4, which workdirectly with the Gagliardo semi-norm for fractional Sobolev spaces. Proof of Theorem 4.1. Assuming f and R (see (4.2)) satisfy hypotheses (2.9) and (2.11), weprove the theorem for a solution of the CHR model in dimension N = 3 . The proof in dimension N = 2 follows as in Theorem 1.5. The result for the truncated CHR model follows from asimplified version of the following argument.Let c be a strong solution of the CHR model (4.1) on Ω T for 0 < T ≤ . For convenience, wewill define R c := R ( c, ∆ c ) . Note that R c ∈ H / , / (Σ T ) (see Theorem 2.14) and by Corollary2.8 may be extended to ˜ R c ∈ H / , / (Σ ) satisfying the bound k ˜ R c k H / , / (Σ ) ≤ C (cid:16) (1 + T − / ) kR c k H , (Σ T ) + kR c k H / , / (Σ T ) (cid:17) . Let F be the extension by 0 of ∆ f ′ ( c ) ∈ H , (Ω T ) to H , (Ω ) . 30e consider the PDE for ¯ c on the extended domain Ω : ∂ t ¯ c + ∆ ¯ c = F in Ω ,∂ ν ¯ c = 0 on Σ ,∂ ν (∆¯ c ) = ˜ R c on Σ , ¯ c (0) = c in Ω . (4.14)If problem (4.14) admits a solution ¯ c ∈ H , (Ω), then as (4.14) coincides with the CHR model(4.1) on Ω T , by uniqueness (see Theorem 6.7), ¯ c | Ω T = c. We may apply Theorem 6.7 (with k = 0)to conclude that (4.14) admits a unique solution ¯ c satisfying the slightly modified bound: k ¯ c k H , (Ω ) + k∇ ¯ c k L ∞ (0 , L (Ω)) ≤ C (cid:16) k ∆ c k L (Ω) + k F k H , (Ω ) + k ˜ R c k H / , / (Σ ) (cid:17) ≤ C (cid:16) k ∆ c k L (Ω) + k ∆ f ′ ( c ) k H , (Ω T ) + (1 + T − / ) kR c k H , (Σ T ) + kR c k H / , / (Σ T ) (cid:17) =: C ( k ∆ c k L (Ω) + A + A + A ) . (4.15)We note that C in the above estimate is independent of T , and the extension Ω T to Ω wasspecifically done to control dependence of constants on T in the above expression. We nowestimate each term in the above expression. Term A : Recalling the proof of Theorem 1.5 (see (4.12)), we have I [ c ( t )] ≤ I [ c ] + T C H ( ∂ Ω) =: ¯ η ( c , T )for all t ∈ (0 , T ) , implying k∇ c k L ∞ (0 ,T ; L (Ω)) ≤ ¯ η ( c , T ) . (4.16)We have ∆ f ′ ( c ) = f ′′′ ( c ) k∇ c k + f ′′ ( c )∆ c. (4.17)Making use of the Gagliardo-Nirenberg inequality (Theorem 2.5), we control the quadratic termof the previous function: k∇ c k L (Ω) ≤ C (Ω) k∇ c k / H (Ω) k∇ c k / L (Ω) . Consequently for ǫ > , (4.17) is bounded in L as k ∆ f ′ ( c ) k L (Ω) ≤ C ( f )( k∇ c k L (Ω) + k ∆ c k L (Ω) ) ≤ C ( f, Ω)(¯ η / k∇ c k / H (Ω) + k ∆ c k L (Ω) ) ≤ C ( f, Ω) (cid:18) ǫ k∇ c k H (Ω) + 1 ǫ ¯ η + k ∆ c k L (Ω) (cid:19) , where we used (2.8), (4.16), and the previous inequality. Taking the L norm over (0 , T ) , wefind k ∆ f ′ ( c ) k H , (Ω T ) ≤ C ( f, Ω) (cid:18) ǫ k∇ c k H , (Ω T ) + 1 ǫ √ T ¯ η + √ T k ∆ c k L ∞ (0 ,T ; L (Ω)) (cid:19) . (4.18) Term A : We note that H (Ω) = [ H (Ω) , H (Ω)] / , so k · k H (Ω) ≤ C (Ω) k · k / H (Ω) k · k / H (Ω) forsome constant C (Ω) > k c k H , (Ω T ) ≤ C (Ω) Z T k c ( t ) k H (Ω) k c ( t ) k H (Ω) dt ≤ C (Ω) k c k H , (Ω T ) k c k H , (Ω T ) ≤ C (Ω) √ T k c k L ∞ (0 ,T ; H (Ω)) k c k H , (Ω T ) ≤ C (Ω) √ T (cid:16) k c k L ∞ (0 ,T ; H (Ω)) + k c k H , (Ω T ) (cid:17) . (4.19)31sing this computation, Remark 2.3, and the continuity of the trace in H (Ω), we estimate Term A as (1 + T − / ) kR c k H , (Σ T ) ≤ C (Ω)(1 + T − / ) kR c k H , (Ω T ) ≤ C ( R , Ω)(1 + T − / ) (cid:16) k c k H , (Ω T ) + |R (0 , |√ T (cid:17) ≤ C ( R , Ω) T / (cid:16) k c k L ∞ (0 ,T ; H (Ω)) + k c k H , (Ω T ) + T / (cid:17) . (4.20) Term A : Since k · k H / (Γ) ≤ C k · k H (Ω) , it follows kR c k H / , (Σ T ) can be estimated by the samemethod as term A , so we restrict our attention to the semi-norm |R c | H , / (Σ T ) . Setting ( R c ) T ( x, t ) := R c ( x, T t ) and c T ( x, t ) := c ( x, T t ), using Lemma 2.10, Remark 2.3,Theorem 2.14, and Proposition 2.15, it follows that |R c | H , / (Σ T ) = T / | ( R c ) T | H , / (Σ ) = T / |R ( c T ) | H , / (Σ ) ≤ C ( R ) T / (cid:0) | c T | H , / (Σ ) + | ∆ c T | H , / (Σ ) (cid:1) ≤ CT / k c T k H , / (Ω ) ≤ CT / ( k c T k H , (Ω ) + | c T | H , / (Ω ) ) . Using a direct change of variables, we have that k c T k H , (Ω ) = T − / k c k H , (Ω T ) . By Proposition2.9 and a change of variables, we have | c T | H , / (Ω ) ≤ C k ∂ t ( c T ) k H , (Ω ) = CT k ( ∂ t c ) T k H , (Ω ) = CT / k ∂ t c k H , (Ω T ) . Consolidating these estimates along with (4.19), we find |R c | H , / (Σ T ) ≤ C ( R , Ω) T / (cid:0) k c k H , (Ω T ) + k c k L ∞ (0 ,T ; H (Ω)) (cid:1) . (4.21)Returning to (4.15) and combining the bounds (4.18), (4.20), and (4.21), we find k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) ≤k ¯ c k H , (Ω ) + k∇ ¯ c k L ∞ (0 , L (Ω)) ≤ C k ∆ c k L (Ω) + C ( f, Ω) (cid:18) ǫ k∇ c k H , (Ω T ) + 1 ǫ √ T ¯ η + √ T k ∆ c k L ∞ (0 ,T ; L (Ω)) (cid:19) + C ( R , Ω) T / (cid:0) k c k H , (Ω T ) + k c k L ∞ (0 ,T ; H (Ω)) + T / (cid:1) . By the coercivity of f (2.9), the definition of ¯ η (see above (4.16)), and the Poincar´e inequality,we have k c k H (Ω) ≤ C (Ω) (cid:18) ¯ η ( c , T ) + 1 δ (cid:19) + k∇ c k L (Ω) , and returning to the above estimate, we have k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) ≤ C k ∆ c k L (Ω) + C ( f, Ω) (cid:18) ǫ k∇ c k H , (Ω T ) + 1 ǫ √ T ¯ η + √ T k ∆ c k L ∞ (0 ,T ; L (Ω)) (cid:19) + C ( R , Ω) T / (cid:18) ¯ η + 1 δ + k∇ c k L ∞ (0 ,T ; L (Ω)) + k c k H , (Ω T ) + T / (cid:19) ≤ C k∇ c k L (Ω) + C ( f, R , Ω) (cid:0) ǫ + T / (cid:1) (cid:0) k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) (cid:1) + C ( R , Ω) T / (cid:18) ¯ η + 1 δ + T / (cid:19) + C ( f, Ω) ǫ √ T ¯ η . < ǫ < / (4 C ( f, R , Ω)) , we have for all 0 < T < (1 / (4 C ( f, R , Ω))) , that k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) ≤ C k ∆ c k L (Ω) + C ( R , Ω) T / (cid:18) ¯ η + 1 δ + T / (cid:19) + C ( f, Ω) ǫ √ T ¯ η =: C k ∆ c k L (Ω) + η ( c , T ) . N = 2 This section is devoted to proving sufficient regularity of a solution to the truncated CHR model(4.4) such that we recover a solution of the CHR model (1.12) (see also (4.1)) with R given by(1.6) and f given by (1.3). We will prove inclusion of a solution of the truncated CHR model in ahigher order anisotropic Sobolev space, at which point we will make use of embedding theoremsto recover continuity of the second derivatives in space and time.We refine the assumptions previously used to prove strong existence: • We assume that the chemical energy density is governed by f ∈ C , ( R ) . (5.1) • For the reaction rate, we assume R ∈ C , ( R ) . (5.2) Remark 5.1. For R and f satisfying (5.2) and (5.1), respectively, and recalling (4.2), the chainrule provides the bound k∇Rk C , ( R ; R ) ≤ C. The proof of the existence as claimed in Theorem 1.8 proceeds in two primary steps. First,a fixed point argument analogous to Theorem 1.5 is applied to obtain existence of a solution tothe truncated CHR model (4.4) belonging to H , / (Ω T ) for sufficiently small T > . Lookingto Remark 5.3, we see that the use of another fixed point argument is driven by necessity–versusbeing able to directly bootstrap from a strong solution to higher regularity. The argument willmake sharp use of the growth given by the exponents of the Gagliardo-Nirenberg inequality(Theorem 2.5), and hence critically relies on the a priori “smallness” estimate provided byTheorem 4.1. Second, we directly bootstrap to show that given appropriate initial conditions,a solution of the truncated CHR model (4.4) belongs to H , / (Ω T ), at which point we maydirectly apply embedding theorems to conclude existence of the desired solution to the CHRmodel (4.1). Given that our argument places restrictions on the class of admissible initialconditions, we last show that this class of functions is non-empty. Theorem 5.2. Suppose Ω ⊂ R is an open bounded set with smooth boundary, and f and R (see (4.2)) satisfy assumptions (5.1) and (5.2). There is λ ( R , Ω) > such that if c ∈ H (Ω) such that ∂ ν c = 0 and k∇ c k L (Ω) ≤ λ , then there is T > such that a solution of the truncatedCHR model (4.4) exists on the interval Ω T satisfying the estimate k c k H , / (Ω) ≤ C ( f, c , R , Ω , T ) . roof. We apply Schaefer’s fixed point theorem [29] to obtain existence. Define the operator A : v c by the PDE ∂ t c + ∆ c = (∆ f ′ ) α ( v ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν (∆ c ) = R ( v, ∆ v ) on Σ T ,c (0) = c in Ω . (5.3)We choose the domain of A such that A is both compact and range(A) ⊂ H , / (Ω T ). Definethe Banach space B := H , (Ω T ) ∩ L (0 , T ; W ,r (Ω)) ∩ L ∞ (0 , T ; W ,r (Ω))equipped with the sum of norms, for r yet to be determined. We claim H , / (Ω T ) ֒ → ֒ → B . As H , / (Ω T ) ֒ → ֒ → [ H , (Ω T ) , H , / (Ω T )] θ for θ ∈ (0 , 1) (see, e.g., Exercise 16.26 of [44] and [46]), it suffices to show that[ H , (Ω T ) , H , / (Ω T )] θ = H θ ,θ (5 / (Ω T ) ֒ → B (5.4)for some θ ∈ (0 , H θ ,θ (5 / (Ω T ) ֒ → H , (Ω T ) for θ sufficiently close to 1. Usinga Besov embedding theorem (see Theorem 17.51 of [44]), we have H θ (Ω) ֒ → W ,r (Ω) forall r < − θ )5 . It immediately follows that H θ ,θ (5 / (Ω T ) ֒ → L (0 , T ; W ,r (Ω)) . To concludethe last embedding necessary to prove (5.4) we make use of a trace theorem (see Theorem3.1 of [45]), which shows that c ∈ H θ ,θ (5 / (Ω T ) is continuously embedded in the space of BU C (0 , T ; [ H θ (Ω) , L (Ω)] / ( θ ) . As [ H θ (Ω) , L (Ω)] / ( θ = H θ − (Ω) by Proposition 2.13, wemay once again make use of a Besov embedding theorem [44] to conclude for any r ≥ , there is θ sufficiently close to 1 such that H θ ,θ (5 / (Ω T ) ֒ → L ∞ (0 , T ; W ,r (Ω)) . This concludes the claim.We now prove that the hypotheses of Schaefer’s fixed point theorem are satisfied by the oper-ator A : B → B for appropriate initial data c ; these are summarily referred to as Compactness,Continuity, and Boundedness (see the proof of Theorem 1.5). This will complete the proof. Compactness: We estimate R ( v, ∆ v ) and (∆ f ′ ) α ( v ) in the norms for H / , / (Ω T ) and H , / (Ω T ) respectively. Hence we define β v := R ( v, ∆ v ) . By continuity of the trace map (seeTheorem 2.14) and Remarks 2.3 and 5.1, we have k β v k H / , / (Σ T ) ≤ C (Ω , T ) k β v k H , / (Ω T ) ≤ C ( R , Ω , T )( k∇ β v k H , (Ω T ) + k v k H , (Ω T ) + 1) , (5.5)For simplicity, we show how to control the higher order spatial derivatives of β v by looking attwo derivatives in the same direction (mixed derivatives are similar): ∂ i β v =( ∂ s R ( v, ∆ v )( ∂ i v ) + 2( ∂ s ∂ w R )( v, ∆ v ) ∂ i v∂ i ∆ v + ( ∂ s R )( v, ∆ v ) ∂ i v + ( ∂ w R )( v, ∆ v )( ∂ i ∆ v ) + ( ∂ w R )( v, ∆ v ) ∂ i ∆ v. (5.6)Given Remark 5.1, the terms in (5.6) are controlled in H , (Ω T ) by the norm of v in H , (Ω T )plus C ( R ) k ( ∂ i ∆ v ) k H , (Ω T ) . To control this term, we make use of the Gagliardo-Nirenberginequality (Theorem 2.5): k ∂ i ∆ v k L (Ω) ≤ C ( k∇ ∆ v k aL r (Ω) k ∆ v k − aL r (Ω) + k ∆ v k L r (Ω) ) , (5.7)34here a = 1 / /r. Consequently, choosing r ≥ , we have k ( ∂ i ∆ v ) k H , (Ω T ) = (cid:18)Z T k ∂ i ∆ v k L (Ω) dt (cid:19) / ≤ C k ∆ v k − a ) L ∞ (0 ,T ; L r (Ω)) (cid:18)Z T k∇ ∆ v k aL r (Ω) dt (cid:19) / + C k ∆ v k L ∞ (0 ,T ; L r (Ω)) ≤ C k ∆ v k − a ) L ∞ (0 ,T ; L r (Ω)) (cid:18)Z T k∇ ∆ v k L r (Ω) dt (cid:19) / + C k ∆ v k − a )+1 L ∞ (0 ,T ; L r (Ω)) + C ≤ C k v k − a )+1 B + C, (5.8)where we used the inequality s ≤ C ( s − a )+1 + 1) since a ≤ / . Thus, using the definition ofthe space B , (5.5), and (5.8), we have k β v k H / , / (Σ T ) ≤ C k v k − a )+1 B + C. Due to the truncated Laplacian (4.3), estimation of the bulk term (∆ f ′ ) α ( v ) in H , (Ω T ) isstraightforward. By Theorem 6.7 (with k = 1), we conclude k c k H , / (Ω T ) ≤ C k v k − a )+1 B + C, which implies A : B → B is compact by the claim regarding (5.4). Continuity: Suppose v n → v in B . To show that A [ v n ] → A [ v ], by Theorem 6.7 (with k = 1)and the claim preceding (5.4), it is sufficient to show that the data converges as follows:(∆ f ′ ) α ( v n ) → (∆ f ′ ) α ( v ) in H , / (Ω T ) ,β v n := R ( v n , ∆ v n ) → R ( v, ∆ v ) = β v in H , / (Ω T ) , where we have used Theorem 2.14 to reduce our consideration to convergence on Ω T versus Σ T . We focus our attention on the second convergence, the first being similar.Up to a subsequence, we may assume ∇ k v n → ∇ k v a.e. in Ω T for k ∈ { , . . . , } . To see that β v n → β v in H , (Ω T ) , recall Remark 2.3 to find k β v n − β v k H , (Ω T ) = Z T Z Ω |R ( v n , ∆ v n ) − R ( v, ∆ v ) | dx dt ≤ C ( R ) Z T Z Ω (cid:0) | v n − v | + | ∆ v n − ∆ v | (cid:1) dx dt. Thus by the convergence of v n in B , we directly have convergence in H , (Ω T ) . To prove conver-gence in H , / (Ω T ) we argue using Remark 2.3 and the Galiardo type semi-norm (see (2.24)): | β v n − β v | H , / (Ω T ) = Z Ω Z T Z T |R ( v n , ∆ v n )( t ) − R ( v, ∆ v )( s ) | | t − s | dt ds dx ≤ C ( R ) Z Ω Z T Z T | v n ( t ) − v ( s ) | | t − s | + | ∆ v n ( t ) − ∆ v ( s ) | | t − s | dt ds dx = C ( R ) (cid:16) | v n − v | H , / (Ω T ) + | ∆ v n − ∆ v | H , / (Ω T ) (cid:17) . As ( v n , ∆ v i ) → ( v, ∆ v ) in [ H , / (Ω T )] (see Proposition 2.15), we are done. Convergence of firstorder derivatives in space is done similarly. 35o show that the second order derivatives converge is more involved. We show convergencefor repeated derivatives as in (5.6), with mixed derivatives being similar. We explicitly showconvergence of the term ( ∂ w R )( v n , ∆ v n )( ∂ i ∆ v n ) with the remaining terms being simpler. De-composing the difference of products, for t − a.e. we compute k ( ∂ w R )( v n , ∆ v n )( ∂ i ∆ v n ) − ( ∂ w R )( v, ∆ v )( ∂ i ∆ v ) k L (Ω) ≤k ( ∂ w R )( v n , ∆ v n ) (cid:2) ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) (cid:3) k L (Ω) + k (cid:2) ( ∂ w R )( v n , ∆ v n ) − ( ∂ w R )( v, ∆ v ) (cid:3) ( ∂ i ∆ v ) k L (Ω) ≤ C ( R ) k ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) k L (Ω) + k (cid:2) ( ∂ w R )( v n , ∆ v n ) − ( ∂ w R )( v, ∆ v ) (cid:3) ( ∂ i ∆ v ) k L (Ω) , (5.9)where we used Remark 5.1. Up to another subsequence of n, for t -a.e., the second term goes to0 by the Lebesgue dominated convergence theorem. Taking another subsequence if necessary,we apply H¨older’s inequality and the Sobolev-Gagliardo-Nirenberg embedding theorem to showthat the first term also goes to 0 for t -a.e.: k ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) k L (Ω) ≤ k ∂ i ∆ v n − ∂ i ∆ v k L (Ω) k ∂ i ∆ v n + ∂ i ∆ v k L (Ω) ≤ C Ω k∇ ∆( v n − v ) k H (Ω) k∇ ∆( v n + v ) k H (Ω) → · k∇ ∆ v k H (Ω) = 0 . We now apply the generalized Lebesgue dominated convergence theorem to prove ( ∂ i ∆ v n ) → ( ∂ i ∆ v ) in H , (Ω T ): k ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) k H , (Ω T ) = Z T k ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) k L (Ω) dt. (5.10)We bound the integrand pointwise for t -a.e. using estimate (5.7) and that k v n k B → k v k B in B : k ( ∂ i ∆ v n ) − ( ∂ i ∆ v ) k L (Ω) ≤ C (cid:16) k ∂ i ∆ v n k L (Ω) + k ∂ i ∆ v k L (Ω) (cid:17) ≤ C (cid:16) k ∆ v n k aW ,r (Ω) k ∆ v n k − a ) L r (Ω) + k ∆ v k aW ,r (Ω) k ∆ v k − a ) L r (Ω) (cid:17) ≤ C sup n n k ∆ v n k − a ) L ∞ (0 ,T ; L r (Ω)) o (cid:16) k ∆ v n k W ,r (Ω) + k ∆ v k W ,r (Ω) + 1 (cid:17) ≤ C sup n n k v n k − a ) B o (cid:16) k ∆ v n k W ,r (Ω) + k ∆ v k W ,r (Ω) + 1 (cid:17) ∈ L (0 , T ) . Thus we apply the generalized Lebesgue dominated convergence theorem to conclude (5.10)converges to 0 . Likewise, we conclude that the left-hand side of (5.9) goes to 0 in L (0 , T ), fromwhich we conclude the desired convergence of second order terms, and finally continuity of theoperator A. Boundedness: We show that the set { c ∈ B : c = λA [ c ] for λ ∈ (0 , } is bounded in B for λ ∈ (0 , . We assume λ = 1; the argument is the same for other λ. Thus, suppose c = A [ c ] ∈ H , / (Ω T ) . Making use of the bound (4.5) and assumptions on f ,it straightforward to show that (∆ f ′ ) α ( c ) is bounded in H , / (Ω T ) in terms of k c k H , (Ω T ) ≤ C ( c , Ω , T ). Now, we control β c := R ( c, ∆ c ) in H / , / (Σ T ) . Given Proposition 2.15, the normof β c in H , / (Σ T ) is controlled by k c k H , (Ω T ) ≤ C ( c , Ω , T ). We summarize these initial boundsas k β c k H , / (Σ T ) + k (∆ f ′ ) α ( c ) k H , / (Ω T ) ≤ C ( c , f, R , Ω , T ) . (5.11)36o bound β c in H / , (Σ T ), we first look at ∇ β c in H , (Ω T ) . We control the repeated derivative ∂ i β c , as in (5.6); control of the mixed derivatives is analogous.First, we impose a restriction on the initial condition such that k ∆ c k L (Ω) ≤ k , where k ∈ N is yet to be chosen. By (4.5), for all T > k c k H , (Ω T ) + k∇ c k L ∞ (0 ,T ; L (Ω)) ≤ Ck + η, (5.12)where the constant C > T. Looking to terms arising in (5.6), we square theGagliardo-Nirenberg inequality (Theorem 2.5) in dimension N = 2 and use (5.12) to find for t -a.e. k ∂ i (∆ c ) k L (Ω) ≤ C (Ω) k ∆ c k H (Ω) k ∆ c k L (Ω) ≤ C (Ω) (cid:18) Ck + η (cid:19) k ∆ c k H (Ω) . (5.13)Using H¨older’s inequality, Young’s inequality, the Sobolev-Gagliardo-Nirenberg embedding the-orem, and a trace theorem (Theorem 3.1 of [45]), we have for t -a.e. k ∂ i c ∂ i (∆ c ) k L (Ω) ≤ (cid:16) C (Ω) k c k L ∞ (0 ,T ; H (Ω)) + k ∂ i (∆ c ) k L (Ω) (cid:17) ≤ (cid:16) C (Ω , T ) k c k H , (Ω T ) + k ∂ i (∆ c ) k L (Ω) (cid:17) . (5.14)Recalling (5.2) and noting the terms in (5.6), we see that the bounds (5.13), (5.14), and k c k H , (Ω T ) ≤ C (from (4.5)) imply k ∂ i β c k H , (Ω T ) ≤ C ( c , f, R , Ω) (cid:18) C ( T ) + (cid:18) Ck + η (cid:19) k ∆ c k H , (Ω T ) (cid:19) . (5.15)As noted previously, an argument analogous to the above succeeds in controlling the full deriva-tive ∇ β c . Furthermore, as R is Lipschitz (see Remark 5.1), it is direct to conclude β c iscontrolled in H , (Ω T ) by k c k H , (Ω T ) . Thus by (5.15) and the trace inequality k · k H / (Γ) ≤ C (Ω) k · k H (Ω) [44], we have k β c k H / , (Σ T ) ≤ C ( c , f, R , Ω) (cid:18) C ( T ) + (cid:18) Ck + η (cid:19) k ∆ c k H , (Ω T ) (cid:19) . (5.16)Lastly, we will use the method established in the proof of Theorem 4.1 to extend the bulkand boundary data to Ω and Σ , and then apply Theorem 6.7 (with k = 1) to bound c ∈ H , / (Ω T ) by k c k H , / (Ω T ) ≤ C ( c , f, R , Ω , T ) + C ( c , f, R , Ω) (cid:18) Ck + η (cid:19) k ∆ c k H , (Ω T ) . (5.17)Supposing we have estimate (5.17), choosing T > k ∈ N large enoughsuch that C ( c , f, R , Ω) (cid:18) Ck + η (cid:19) < , we may directly conclude the proof of boundedness. So it only remains to prove (5.17).We use Corollary 2.8 to find an extension ˜ β c ∈ H / , / (Ω ) of β c such that k ˜ β c k H / , / (Σ ) ≤ C (cid:0) (1 + T − / ) k β c k H , (Σ T ) + k β c k H / , / (Σ T ) (cid:1) . f ∈ H , / of (∆ f ′ ) α ( c ) such that k ˜ f k H , / (Ω ) ≤ C (cid:0) (1 + T − / ) k (∆ f ′ ) α ( c ) k H , (Ω T ) + k (∆ f ′ ) α ( c ) k H , / (Ω T ) (cid:1) . With this in hand, we consider the PDE for ˜ c ∂ t ˜ c + ∆ ˜ c = ˜ f in Ω ,∂ ν ˜ c = 0 on Σ ,∂ ν (∆˜ c ) = ˜ β c on Σ , ˜ c (0) = c in Ω . Note, by uniqueness (see Theorem 6.7), ˜ c | Ω T = c. Theorem 6.7 (with k = 1), bound (5.11), and(5.16) then show k c k H , / (Ω T ) ≤ k ˜ c k H , / (Ω ) ≤ C (Ω , k ˜ f k H , / (Ω ) + k ˜ β c k H / , / (Σ ) ) ≤ C ( c , f, R , Ω , T ) + C ( c , f, R , Ω) (cid:18) Ck + η (cid:19) k ∆ c k H , (Ω T ) . Note the first constant may blow up as T → , and the extensions in time have been used toguarantee this does not happen to the coefficient of k . Remark 5.3. In the previous theorem, we were unable to directly bootstrap to higher regularityand had to re-do the fixed point argument first used to gain a strong solution. This is because c ∈ H , (Ω T ) is not sufficient to guarantee R ( c, ∆ c ) ∈ H / , / (Σ T ) . However, we will see that c ∈ H , / (Ω T ) is sufficient to guarantee R ( c, ∆ c ) ∈ H / , / (Σ T ) . We now prove the next step in regularity, inclusion of the solution in H , / (Ω T ) . Theorem 5.4. Suppose Ω ⊂ R is an open bounded smooth domain. Further suppose f and R satisfy (5.1) and (5.2). There is λ ( R , Ω) > such that if c ∈ H (Ω) such that ∂ ν c = 0 ,∂ ν (∆ c ) = R ( c , ∆ c ) and k∇ c k L (Ω) ≤ λ , then there is T > such that a solution of thetruncated CHR model (4.4) exists on the domain Ω T satisfying the estimate k c k H , / (Ω) ≤ C ( f, c , R, Ω , T ) . Proof. Let λ > c ∈ H , / (Ω T ) . We show that R ( c, ∆ c ) ∈ H , / (Ω T ) , which will imply that R ( c, ∆ c ) ∈ H / , / (Σ T ) (see Theorem 2.14). As R is Lipschitz, makinguse of Proposition 2.15 and Theorem 5.2, it is straightforward to show that kR ( c, ∆ c ) k H , / (Ω T ) ≤ C ( R , Ω) (cid:0) k c k H , / (Ω T ) + k ∆ c k H , / (Ω T ) + 1 (cid:1) ≤ C ( f, c , R , Ω , T ) . Thus it remains to bound the third derivative of R ( c, ∆ c ) . Looking to (5.6), we see that thedifficult terms which will need bounded in H , (Ω T ) are ∂ i (∆ c ) ∂ i (∆ c ) and ( ∂ i ∆ c ) ; using Young’sinequality, we reduce this to consideration of the terms ( ∂ i (∆ c )) / and ( ∂ i ∆ c ) . The Gagliardo-Nirenberg inequality (Theorem 2.5) provides the estimates for t -a.e. in (0 , T ) k ∂ i (∆ c ) k L (Ω) ≤ C (Ω) k ∂ i (∆ c ) k / H (Ω) k ∂ i (∆ c ) k / L (Ω) , k ∂ i ∆ c k L (Ω) ≤ C (Ω) k ∂ i (∆ c ) k / H (Ω) k ∂ i (∆ c ) k / L (Ω) , which in turn by Theorem 5.2 and H , / (Ω T ) ֒ → BU C (0 , T ; H (Ω)) [45] shows k ( ∂ i (∆ c )) / k H , (Ω T ) + k ( ∂ i (∆ c )) k H , (Ω T ) ≤ C (Ω)( k∇ c k L ∞ (0 ,T ; L (Ω)) + 1) k c k H , (Ω T ) ≤ C ( f, c , R , Ω , T ) . kR ( c, ∆ c ) k H / , / (Σ T ) + k (∆ f ′ ) α ( c ) k H , / (Ω T ) ≤ C ( f, c , R , Ω , T ) . To apply the regularity Theorem 6.7 (with k = 2), we must make sure that the compatibilitycondition is satisfied (i.e., β (0) = ∂ ν (∆ c )). To see that this is the case, we again note that c ∈ H , / (Ω T ) implies c ∈ BU C (0 , T ; H (Ω)) ֒ → ( c, ∆ c ) ∈ BU C (0 , T ; [ H / (Γ)] ) . Consequently, R ( c, ∆ c )( · , 0) = R ( c ( · , , ∆ c ( · , R ( c , ∆ c ) = ∂ ν (∆ c ) , (5.18)verifying the compatibility condition.To understand the utility of a solution in H , / (Ω T ) , we have the following lemma. Lemma 5.5. H , / (Ω T ) continuously embeds into BU C (0 , T ; H (Ω)) . Proof. This is a consequence of Theorem 3.1 in [45], which holds for noninteger derivatives.We now have sufficient power to prove existence of a solution to the CHR model with expo-nential boundary conditions for sufficiently small intervals of time. Proof of Theorem 1.8. This is proof is mainly a matter of choosing truncations. As c ∈ H (Ω), k c k C (Ω) =: α < ∞ . We can construct functions ˜ R and ˜ f such that ˜ R = R on B (0 , α + 1) and˜ f = f on [ ǫ/ , − ǫ/ 2] and the hypotheses of Theorem 5.4 are satisfied. Consider the PDE ∂ t c + ∆ c = (∆ ˜ f ) α +1 ( c ) in Ω T ,∂ ν c = 0 on Σ T ,∂ ν (∆ c ) = ˜ R ( c, ∆ c ) on Σ T ,c (0) = c in Ω , for which there is a solution c ∈ H , / (Ω T ) for some T > c ∈ C ([0 , T ]; C ,a (Ω)) for some a > 0. By continuity, there is some interval [0 , T ] for which˜ R ( c, ∆ c ) = R ( c, ∆ c ) and (∆ ˜ f ) α +1 ( c ) = ∆ f ( c ), proving the theorem. Remark 5.6. Lastly, we would like to make sure we have proven something which is not triviallytrue. Explicitly, we claim there are initial conditions which satisfy the hypothesis of Theorem1.8. Recall, as in Singh et. al. [58], by (1.6) and (4.2), we have that R ( c, ∆ c ) = − ( R ins − R ext ) = k ext c exp( β ( µ − µ e )) − k ins exp( β ( µ e − µ )) , where all constants k ext , k ins , β, µ e are positive. Consider the case of a constant c ∈ (0 , ,then we have R ( c , ∆ c ) = − R ( c , f ′ ( c )) = k ext c exp( β ( f ′ ( c ) − µ e )) − k ins exp( β ( µ e − f ′ ( c ))) . Since lim z → f ′ ( z ) = −∞ and lim z → f ′ ( z ) = ∞ , it follows that lim c → R ( c , ∆ c ) = −∞ , lim c → R ( c , ∆ c ) = ∞ . By the intermediate value theorem, there is c ∈ (0 , such that R ( c , µ ) = 0 . It then followsthat c is an admissible condition for Theorem 1.8 as k∇ c k L (Ω) = 0 ≤ λ and ∂ ν (∆ c ) = 0 = R ( c , µ ) . Considering sufficiently small perturbations of c , we may find other admissible initialconditions. Remark 5.7. In Theorem 1.8, the method of proof can also account for the case in which theinitial chemical potential is small, i.e. k − ∆ c + f ′ ( c ) k L (Ω) ≤ λ. However, we do not know ifthis class of admissible functions is non-empty. Appendix The purpose of this appendix is to develop a relatively self contained presentation for regularity ofthe PDE (6.1) below. The first two results show how gradient flows provide regularity in the caseof weak data, and then in the case of L data. It was originally our desire to use interpolationtheorems to obtain regularity for the case of intermediate data in the space H k, k/ (Ω T ) . However, this approach demands a little too much of an appendix (see Remark 6.8), and werefer to a classical result of Lions and Magenes [46].We analyze the PDE ∂ t c − div(Λ ∇ µ ) = g in Ω T , − div(Λ ∇ c ) = µ in Ω T , (Λ ∇ c ) · ν = 0 on Σ T , (Λ ∇ µ ) · ν = 0 on Σ T ,c (0) = c in Ω , (6.1)for Λ ∈ C ∞ ( ¯Ω , Pos( N )) with data g and c . Note that λ ( x ) k y k ≤ (Λ( x ) y ) · y ≤ λ N ( x ) k y k , x ∈ ¯Ω , y ∈ R N , where λ ( x ) and λ N ( x ) are the smallest and largest eigenvalues of Λ( x ), respectively. We furtherremark that we only use Λ = I in the previous sections, but including this generality here doesnot create additional complications.Let us now make clear by what we mean by a solution. First, define the Hilbert space V := { w ∈ H (Ω) : − R Ω w = 0 } , with inner product( w, v ) V := Z Ω ( ∇ w, ∇ v ) Λ dx, (6.2)where for vectors x, y ∈ R N ( x, y ) Λ := (Λ x ) · y. Note that k w k V is equivalent to the standard H norm by (6.2) the Poincar´e inequality. For anyelement L ∈ V ∗ , by the Riesz representation theorem, we have that h L, w i V ∗ ,V = R Ω ( ∇ z L , ∇ w ) Λ for some z L ∈ V ; i.e., z L is a weak solution of the Neumann problem ( − div(Λ ∇ w ) = L in Ω , Λ ∇ w · ν = 0 on Γ . (6.3)Furthermore, k z L k V = k L k V ∗ . (6.4)We also have the following result on regularity of z L . This is a more general case of the estimatein Theorem 2.4. Lemma 6.1. [39] Let Ω be an open, bounded set with C k +2 boundary. Suppose L ∈ H k (Ω) with R Ω L dx = 0 , Λ ∈ C ∞ ( ¯Ω , Pos ( N )) , and z L is a weak solution of the Neumann problem (6.3).Then k z L k H k +2 (Ω) ≤ C k L k H k (Ω) . Assuming g ∈ L (0 , T ; V ∗ ) , we may define z g ∈ L (0 , T ; V ) pointwise in t by the aforemen-tioned isomorphism. Consequently, we may rewrite (6.1) as ∂ t c − div(Λ ∇ µ ) = 0 in Ω T , − div(Λ ∇ c ) − z g = µ in Ω T , (Λ ∇ c ) · ν = 0 on Σ T , (Λ ∇ µ ) · ν = 0 on Σ T ,c (0) = c on Ω . Definition. We say that c is a weak solution of (6.1) in Ω T if c ∈ L (0 , T ; H (Ω) ∩ V ) ∩ C ([0 , T ) , L (Ω)) ,∂ t c ∈ L (0 , T ; V ∗ ) ,c (0) = c ∈ V, (6.5) and for t -a.e. and ξ ∈ V , h ∂ t c ( t ) , ξ i V ∗ ,V + Z Ω ( ∇ µ ( t ) , ∇ ξ ) Λ dx = 0 , where for t -a.e. µ ( t ) ∈ H (Ω) ⊂ L (Ω) is defined by duality as ( µ ( t ) , ξ ) L (Ω) := Z Ω (( ∇ c ( t ) , ∇ ξ ) Λ − z g ( t ) ξ ) dx, (6.6) which holds for all ξ ∈ H (Ω) . We proceed in stages of increasing regularity of the data. Theorem 6.2. Let Ω ⊂ R N be an open, bounded set with boundary of class C . Let T > , Λ ∈ C ∞ ( ¯Ω , Pos ( N )) , g ∈ L (0 , T ; V ∗ ) , and c ∈ V. Then there exists a unique weak solution c to (6.1) satisfying the following bound k c k L (0 ,T ; H (Ω)) + k ∂ t c k L (0 ,T ; V ∗ ) ≤ C (cid:0) k c k V + k g k L (0 ,T ; V ∗ ) (cid:1) for some constant C = C (Λ , Ω) > . Proof. Step 1: Minimizing movements scheme. We construct a solution via the method ofminimizing movements. For n ∈ N , we partition our time interval (0 , T ) into n equal size stepsof length τ = T /n and recursively define the finite sequence { c iτ } ni =0 in V as follows: c iτ = argmin c ∈ V (cid:26) τ k c − c i − τ k V ∗ + Z Ω (cid:18) h∇ c, ∇ c i Λ − z g iτ c (cid:19) dx (cid:27) , (6.7)where c τ := c , g iτ := − Z iτ ( i − τ g ( t ) dt, (6.8)and we have implicitly used the embedding of V in L (Ω) ⊂ V ∗ . Note that a minimizer exists asthe functional being minimized is coercive and lower semicontinuous with respect to the weaktopology of V . To see the lower semicontinuity, note that k · k V ∗ is lower semi-continuous withrespect to the weak topology of V as the inclusion i : V → L (Ω) is compact and k · k V ∗ iscontinuous with respect to the strong topology on L (Ω). Step 2: “Discrete” Euler-Lagrange equations. We now compute the “discrete” Euler-Lagrange Equations associated with the minimzation problem (6.7). Since ξ z ξ is linear,using (6.2), (6.3), and (6.4), we can compute the Frechet derivative of the norm in V ∗ :lim t → k v + tw k V ∗ − k v k V ∗ t = lim t → t Z Ω (cid:16) ( ∇ z v + tw , ∇ z v + tw ) Λ − ( ∇ z v , ∇ z v ) Λ (cid:17) dx = lim t → Z Ω (cid:16) ( ∇ z w , ∇ z v ) Λ + t ∇ z w , ∇ z w ) Λ (cid:17) dx = Z Ω ( ∇ z w , ∇ z v ) Λ dx = h v, z w i V ∗ ,V . τ h c iτ − c i − τ , z w i V ∗ ,V + Z Ω (cid:0) ( ∇ c iτ , ∇ w ) Λ − z g iτ w (cid:1) dx = 0 , (6.9)which holds for all w ∈ V . Step 3: Energy estimates and convergence. Letting w = c iτ − c i − τ be the test function in(6.9), we find that1 τ k c iτ − c i − τ k V ∗ + Z Ω (cid:0) ( ∇ c iτ , ∇ ( c iτ − c i − τ )) Λ − z g iτ ( c iτ − c i − τ ) (cid:1) dx = 0 . But using the representation of c iτ − c i − τ in V ∗ , we rewrite this as1 τ k c iτ − c i − τ k V ∗ + Z Ω (cid:16) ( ∇ c iτ , ∇ ( c iτ − c i − τ )) Λ − ( ∇ z g iτ , ∇ z c iτ − c i − τ ) Λ (cid:17) dx = 0 . Applying the Cauchy-Schwarz inequality, we compute Z Ω ( ∇ z g iτ , ∇ z c iτ − c i − τ ) Λ dx ≤ k z g iτ k V k z c iτ − c i − τ k V = k g iτ k V ∗ k c iτ − c i − τ k V ∗ ≤ τ k g iτ k V ∗ + 12 τ k c iτ − c i − τ k V ∗ . Similarly, Z Ω ( ∇ c iτ , ∇ c i − τ ) Λ dx ≤ Z Ω ( ∇ c iτ , ∇ c iτ ) Λ dx + 12 Z Ω ( ∇ c i − τ , ∇ c i − τ ) Λ dx. With this, we obtain the following energy estimate:12 τ k c iτ − c i − τ k V ∗ + 12 Z Ω ( ∇ c iτ , ∇ c iτ ) Λ dx ≤ τ k g iτ k V ∗ + 12 Z Ω ( ∇ c i − τ , ∇ c i − τ ) Λ dx. (6.10)Let ˆ c τ is the linear interpolant and c τ is the left continuous step function associated with thesequence { c iτ } ni =0 as in (3.14) and (3.12), respectively. Likewise we define g τ to be the leftcontinuous step function, g τ ( t ) := g iτ t ∈ (( i − τ, iτ ] , i = 0 , . . . , n − . (6.11)Let k be a positive integer less than or equal to n. From (6.8) and Jensen’s inequality, we have k g τ k L (0 ,kτ ; V ∗ ) = k X i =1 τ (cid:13)(cid:13)(cid:13)(cid:13) − Z iτ ( i − τ g ( t ) dt (cid:13)(cid:13)(cid:13)(cid:13) V ∗ ≤ k X i =1 Z iτ ( i − τ k g ( t ) k V ∗ dt = k g k L (0 ,kτ ; V ∗ ) . (6.12)Using the above bound, recalling (6.2), and that ∂ t ˆ c τ ( t ) = c iτ − c i − τ τ for t ∈ (( i − τ, iτ ), we suminequality (6.10) over i = 1 , . . . , k to find k ∂ t ˆ c τ k L (0 ,kτ ; V ∗ ) + k c kτ k V ≤ k g k L (0 ,kτ ; V ∗ ) + k c k V . This bound implies the control k ∂ t ˆ c τ k L (0 ,T ; V ∗ ) + k c τ k L ∞ (0 ,T ; V ) ≤ k g k L (0 ,T ; V ∗ ) + k c k V . (6.13)42e apply the Aubin-Lions-Simon compactness theorem [57], with the evolution triple ( V, L , V ∗ ) , to find that ˆ c τ converges in L (0 , T ; L (Ω)) as τ → c ∈ L (0 , T ; V ) . We would also liketo consider convergence of the left continuous step functions c τ and hence compute the difference k ˆ c τ ( t ) − ˆ c τ ( t ) k V ∗ ≤ Z t t k ∂ t ˆ c τ k V ∗ dt ≤ C ( t − t ) / . By Corollary 2.6 (which holds with H (Ω) ∗ replaced by V ∗ ) and (6.13), k ˆ c τ ( t ) − ˆ c τ ( t ) k L (Ω) ≤ C (cid:0) k∇ (ˆ c τ ( t ) − ˆ c τ ( t )) k / L (Ω) k ˆ c τ ( t ) − ˆ c τ ( t ) k / V ∗ + k ˆ c τ ( t ) − ˆ c τ ( t ) k V ∗ (cid:1) ≤ C max { ( t − t ) / , ( t − t ) / } , (6.14)which implies convergence of c τ to c in L (0 , T ; L (Ω)) by a direct application of the triangleinequality, and consequently c τ ⇀ c in L (0 , T ; V ). Lastly, we note it is clear that ∂ t ˆ c τ ⇀ ∂ t c and (6.13) holds with ˆ c τ and c τ replaced by c. Step 4: c is the desired solution. Now, let { w k } k ∈ N be dense in V. We then integrate the“discrete” Euler-Lagrange equation (6.9) in time to find Z t t h ∂ t c τ , z w k i V ∗ ,V dt + Z t t (cid:18)Z Ω (cid:0) ( ∇ c − τ , ∇ w k ) Λ − z g τ w k (cid:1) dx (cid:19) dt = 0 . (6.15)We would like to show that z g τ → z g in L (0 , T ; L (Ω)) . By the Lebesgue differentiationtheorem [33], g τ → g in V ∗ for t -a.e. in (0 , T ). Using (6.12), Fatou’s lemma, and the uniformconvexity of a Hilbert space [13], we conclude g τ → g in L (0 , T ; V ∗ ). Consequently, k z g τ − z g k L (0 ,T ; H (Ω)) ≤ C k g τ − g k L (0 ,T ; V ∗ ) → , where we have used (6.4) and linearity of z .Passing τ → Z t t h ∂ t c, z w k i V ∗ ,V dt + Z t t (cid:18)Z Ω (( ∇ c, ∇ w k ) Λ − z g w k ) dx (cid:19) dt = 0for all t , t ∈ [0 , T ] . Using Lebesgue points, we find for all k ∈ N and t -a.e. − h ∂ t c, z w k i V ∗ ,V = Z Ω (( ∇ c, ∇ w k ) Λ − z g w k ) dx. (6.16)By density the above equation holds for all w ∈ V. Using duality, this implies − Z Ω wz ∂ t c dx = − h w, z ∂ t c i V ∗ ,V = −h ∂ t c, z w i V ∗ ,V = Z Ω (( ∇ c, ∇ w ) Λ − z g w ) dx. (6.17)Since ∂ t c ∈ L (0 , T ; V ∗ ) by (6.13), the function z ∂ t c belongs to L (0 , T ; H (Ω)) by (6.3) and(6.4). Similarly z g ∈ L (0 , T ; H (Ω)); hence Lemma 6.1 and (6.17) imply c ∈ L (0 , T ; H (Ω)).Furthermore, if we define µ := − z ∂ t c , by (6.17) we have that Z Ω (( ∇ c, ∇ w ) Λ − z g w ) dx = Z Ω µw dx, (6.18)and by (6.3), Z Ω ( ∇ µ, ∇ w ) Λ dx = − Z Ω ( ∇ z ∂ t c , ∇ w ) Λ dx = h ∂ t c, w i V ∗ ,V . (6.19)43oting that the initial condition c (0) = c and continuity are consequences of (6.14), we have c is a weak solution of (6.1). Step 5: Uniqueness. This follows from an energy argument as in the proof of uniqueness fora solution of the heat equation. Suppose c and c solve (6.1). Then c := c − c solves (6.1)with 0 data. Testing the weak formulation of c with w = c, for t -a.e. we have h ∂ t c, c i V ∗ ,V = Z Ω ( ∇ (div(Λ c )) , ∇ c ) Λ dx, which Theorem II.5.12 of [12] (which shows 2 h ∂ t c, c i V ∗ ,V = ∂ t k c k L (Ω) ) and integrating by partsimply ∂ t (cid:18) k c k L (Ω) (cid:19) + Z Ω (div(Λ c )) dx = 0 , for t -a.e. Thus k c k L (Ω) satisfies the differential inequality ∂ t ( k c k L (Ω) ) ≤ k c (0) k L (Ω) = 0 , which implies c = 0 as desired.We would now like to extend the above analysis to more regular data, and do so in thefollowing theorem. Theorem 6.3. Let Ω be an open, bounded set with C boundary. Let T > , Λ ∈ C ∞ ( ¯Ω , Pos ( N )) , g ∈ L (0 , T ; L (Ω)) , and c ∈ H (Ω) with (Λ ∇ c ) · ν = 0 on Γ . Then there exists a unique solu-tion to (6.1) given by c ∈ H , (Ω T ) satisfying the following bound k c k H , (Ω T ) + k c k L ∞ (0 ,T ; H (Ω)) ≤ C (cid:0) k c k H (Ω) + k g k L (0 ,T ; L (Ω)) (cid:1) (6.20) for some constant C = C (Λ , Ω) > . Proof. Note that up to a shift by a constant function, we can assume that c ∈ V. We highlightthe parts of the analysis differing from the argument in the proof of Theorem 6.2. Makinguse of the Neumann boundary conditions, we integrate by parts the “discrete” Euler-Lagrangeequations (see also (6.18) and (6.19)) derived in the previous proof and use a density argumentto conclude for all w ∈ V and t -a.e. h ∂ t ˆ c τ , w i V ∗ ,V = Z Ω (cid:0) ∇ (div(Λ ∇ c − τ ) + z g τ ) , ∇ w (cid:1) Λ dx. We compute for t ∈ (( i − τ, iτ ). Letting w = ∂ t ˆ c τ ( t ) = c iτ − c i − τ τ and recalling properties of theisomorphism z L for L ∈ L (Ω) provided by (6.3), we have k ∂ t ˆ c τ k L (Ω) = Z Ω (cid:18) ∇ (div(Λ ∇ c iτ ) + z g τ ) , ∇ (cid:16) c iτ − c i − τ τ (cid:17)(cid:19) Λ dx = Z Ω (cid:18) − div(Λ ∇ c iτ )div (cid:16) Λ (cid:16) c iτ − c i − τ τ (cid:17)(cid:17) − div(Λ ∇ z g τ )( ∂ t ˆ c τ ) (cid:19) dx ≤ Z Ω − div(Λ ∇ c iτ )div (cid:16) Λ (cid:16) c iτ − c i − τ τ (cid:17)(cid:17) dx + 12 k div(Λ ∇ z g τ ) k L (Ω) + 12 k ∂ t ˆ c τ k L (Ω) ≤ Z Ω − div(Λ ∇ c iτ )div (cid:16) Λ (cid:16) c iτ − c i − τ τ (cid:17)(cid:17) dx + 12 k g τ k L (Ω) + 12 k ∂ t ˆ c τ k L (Ω) . Note it was in the second equality, when i = 1, that we used the hypothesis (Λ ∇ c ) · ν = 0.Multiplying the previous equation by τ (equivalently integrating in time over (( i − τ, iτ )),utilizing Cauchy’s inequality, and rearranging we find12 k ∂ t ˆ c τ k L (( i − τ,iτ ; L (Ω)) + 12 Z Ω (div(Λ ∇ c iτ )) ≤ Z Ω (div(Λ ∇ c i − τ )) + 12 k g τ k L (( i − τ,iτ ; L (Ω)) i , k ∂ t ˆ c τ k L (0 ,T ; L (Ω)) + k div(Λ ∇ c τ ) k L ∞ (0 ,T ; L (Ω)) ≤k div(Λ ∇ c ) k L (Ω) + k g τ k L (0 ,T ; L (Ω)) ≤k div(Λ ∇ c ) k L (Ω) + k g k L (0 ,T ; L (Ω)) . Thus ∂ t ˆ c τ converges weakly in L (0 , T ; L (Ω)), which allows us to conclude that ∂ t c ∈ L (0 , T ; L (Ω)) by uniqueness of limits. Consequently by Lemma 6.1, z ∂ t c ∈ L (0 , T ; H (Ω))for t -a.e., and we use elliptic regularity once again conclude the bound (6.20). Remark 6.4. Following the above analysis, ∂ t ˆ c τ ∈ L (0 , T ; L (Ω)) for the approximate solutions.Looking at (6.17), we have ˆ c τ ∈ L (0 , T ; H (Ω)) . The norms associated with the aforementionedinclusions are uniformly bounded. Consequently, we may apply the compactness theorem ofAubin-Lions-Simon [57], with H (Ω) ֒ → ֒ → H (Ω) ֒ → L (Ω) , to conclude (up to a subsequence) ˆ c τ → c ∈ L (0 , T ; H (Ω)) . The above results are nearly sufficient to tackle the problems of strong solutions in Section4. It remains to extend to the case of inhomogeneous boundary conditions, but this is achievedwith the aid of liftings from Theorem 2.14. For consideration of regular solutions in Section 5,we will also need results for higher regularity data. We specifically consider (6.1) with Λ = I and inhomogeneous boundary conditions: ∂ t c + ∆ c = g in Ω T ,∂ ν c = α on Σ T ,∂ ν (∆ c ) = β on Σ T ,c (0) = c in Ω . (6.21)For regularity and existence, we have the subsequent theorem. However, let us first make antwo important remarks. Remark 6.5. The following theorem holds for any choice of norms on the anisotropic Sobolevspaces, so long as you are willing to change the constant C (Ω , T ) . In applications within thispaper, it will be important to control exactly how this constant depends on T, so we will oftenextend our considerations to a domain with T = 1 , and control the dependence on T by othermeans. Remark 6.6. We remark that the compatibility conditions for the initial data necessarily arisedue to the embedding H k, k/ (Ω T ) ֒ → BU C (0 , T ; H k (Ω)) . Theorem 6.7. Let Ω ⊂ R N be an open, bounded set with smooth boundary and k ∈ { , , } .Suppose g ∈ H k,k/ (Ω T ) , c ∈ H k +2 (Ω) , α ∈ H µ ,λ (Σ T ) , and β ∈ H µ ,λ (Σ T ) , where µ j and λ j are defined in (2.26) with r = 4 + k and s = 1 + k/ . We further assume the compatibilitycondition ∂ ν c = α ( · , on Γ . If k = 2 , we additionally suppose ∂ ν (∆ c ) = β ( · , on Γ . Thenthere is a unique solution of the PDE (6.21) given by c ∈ H k, k/ (Ω T ) satisfying the bound k c k H k, k/ (Ω T ) ≤ C (Ω , T ) (cid:0) k c k H k +2 (Ω) + k g k H k,k/ (Ω T ) + k α k H µ ,λ (Σ T ) + k β k H µ ,λ (Σ T ) (cid:1) . (6.22) Proof. This theorem is a case of Theorem 5.3 of Chapter 4 in [46]. We only prove the specialcase k = 0 and α = 0, which is the extent of this theorem’s use in Section 4.We reduce to the case of homogeneous boundary conditions by lifting the boundary condition.Looking to Theorem 2.14, this is a matter of writing ∂ ν (∆ w ) in terms of ∂ ν w . Consider smooth γ ∈ C ∞ ( ¯Ω) such that γ = 0 on Γ . Let Pr( x ) := x − d ( x ) ∇ d ( x ) be the (locally well-defined)45rojection of x onto Γ, where d is the signed distance from Γ (negative on the interior of Ω) [10].We have that γ (Pr( x )) = 0 . We then compute the i th derivative by the chain rule:0 = ∂ i ( γ (Pr( x ))) = h∇ γ (Pr( x )) , e i − ∂ i d ( x ) ∇ d ( x ) − d ( x ) ∇ ( ∂ i d ( x )) i . Consequently, ∂ i γ (Pr( x )) = h∇ γ (Pr( x )) , ∂ i d ( x ) ∇ d ( x ) + d ( x ) ∇ ( ∂ i d ( x )) i . We compute the derivative again with respect to the i th direction: h∇ ∂ i γ (Pr( x )) , e i − ∂ i d ( x ) ∇ d ( x ) − d ( x ) ∇ ( ∂ i d ( x )) i = h∇ γ (Pr( x ))( e i − ∂ i d ( x ) ∇ d ( x ) − d ( x ) ∇ ( ∂ i d ( x ))) , ∂ i d ( x ) ∇ d ( x ) + d ( x ) ∇ ( ∂ i d ( x )) i + h∇ γ (Pr( x )) , ∂ i ( ∂ i d ( x ) ∇ d ( x ) + d ( x ) ∇ ( ∂ i d ( x ))) i Choosing x ∈ Γ and recalling ∇ d ( x ) = ν [10] and d ( x ) = 0 , we have ∂ i γ ( x ) = 2 h∇ ∂ i γ ( x ) , ∂ i d ( x ) ∇ d ( x ) i − ( ∂ i d ( x )) h∇ γ ( x ) ∇ d ( x ) , ∇ d ( x ) i + h∇ γ ( x ) , ∂ i d ( x ) ∇ d ( x ) + 2 ∂ i d ( x ) ∇ ( ∂ i d ( x )) i = 2 ∂ i d ( x ) h∇ ∂ i γ ( x ) , ∇ d ( x ) i − ( ∂ i d ( x )) ∂ ν γ ( x )+ h∇ γ ( x ) , ∂ i d ( x ) ∇ d ( x ) + 2 ∂ i d ( x ) ∇ ( ∂ i d ( x )) i . Summing over i , we have∆ γ ( x ) = ∂ ν γ ( x ) + ∂ ν γ ( x ) + 2 h∇ γ ( x ) , ν ∇ ν i = ∂ ν γ ( x ) + ∂ ν γ ( x ) , where we have used that ∇ d ( x ) ∈ ker( ∇ d ( x )) [10]. Consequently, for w satisfying the boundarycondition ∂ ν w = 0, for x ∈ Γ we have that∆( ∂ ν w ) = ∂ ν ( ∂ ν w ( x )) + ∂ ν ( ∂ ν w ( x )) = ∂ ν w ( x ) + ∂ ν w ( x ) . Note we have to be careful as to why ∂ ν ∂ ν = ∂ ν . Then using product rule, we have∆( ∂ ν w ) = ∂ ν ∆ w ( x ) + ∇ w ( x ) : ∇ d ( x ) + ∇ (∆ d )( x ) · ∇ w ( x ) . Consequently, ∂ ν ∆ w ( x ) = ∂ ν w ( x ) + ∂ ν w ( x ) − ∇ w ( x ) : ∇ d ( x ) − ∇ (∆ d )( x ) · ∇ w ( x ) . (6.23)We note this formula holds in the trace sense for any w ∈ H (Ω) as w can be approximatedby smooth functions w i satisfying ∂ ν w i = 0. Such w i can be found by considering the ellipticPDE ∆ w i = f i , ∂ ν w i = 0 where f i belongs to C ∞ ( ¯Ω) and f i → ∆ w in H (Ω). Furthermore, forsmooth w satisfying w = ∂ ν w = ∂ ν w = 0, it follows that ∇ w = 0 and ∇ w = 0 on Γ , so theseequalities also hold in the trace sense for any w ∈ H (Ω) (to approximate such w by smoothfunction with smooth w i satisfying w i = ∂ ν w i = ∂ ν w i = 0, see the space H (Ω) [45]).We then apply Theorem 2.14 to find w ∈ H , (Ω T ) satisfying the bound k w k H , (Ω T ) ≤ C (Ω , T ) k β k H µ ,λ (Σ T ) (6.24)such that w = ∂ ν w = ∂ ν w = 0 and ∂ ν w = β on Γ. By (6.23) and the comment following, wehave ∂ ν ∆ w = β. Considering the trace of w in time, it follows w (0) ∈ H (Ω) with ∂ ν w (0) = 0(see Theorem 3.1 of [45]). Let ¯ c be a strong solution of ∂ t ¯ c + ∆ ¯ c = g − ( ∂ t + ∆ ) w in Ω T ,∂ ν ¯ c = 0 on Σ T ,∂ ν (∆¯ c ) = 0 on Σ T , ¯ c (0) = c − w (0) in Ω , (6.25)as guaranteed by Theorem 6.3. Then c := ¯ c + w solves (6.21), and satisfies the bound (6.22) by(6.24) and (6.20). 46 emark 6.8. If g ∈ H , (Ω T ) and c ∈ H (Ω) , and we wish to conclude c ∈ H , (Ω T ) , wemust impose the compatibility condition ∂ ν (∆ c ) = ∂ ν g ( · , on Γ , which is well defined by atrace theorem of [45]. To approach intermediate regularity via interpolation, we can convenientlydecouple the initial condition from the bulk data. When g = 0 , interpolation of the map c c can be done with the aid of Grisvard’s interpolation results for Sobolev spaces with boundaryconditions defined by normal operators [38]. However, for the map g c , we must have a keenunderstanding of how H , (Ω T ) and { g ∈ H , (Ω T ) : ∂ ν g ( · , 0) = 0 } interpolate. Though we mayintuitively speculate as to what will be the result of this interpolation, such an undertaking isoutside of the scope of an appendix. Acknowledgments This paper is part of the author’s Ph.D. thesis at Carnegie Mellon University under the directionof Irene Fonseca and Giovanni Leoni. 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